Series route: This guide belongs to the complete PSLE Learner’s Guide volume directory. The original PSLE Learning Guide supplies the wider English, Mathematics and Science route, while PSLE Mathematics provide verified subject-specific continuation.
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This volume solves one precise learner problem: the learner writes only p¹⁴ or guesses exponent pairs that add to fifteen, instead of translating the multiplicative partitions 15 and 5×3 into divisor choices. The goal is durable control, not a remembered slogan. A Primary 6 learner should be able to identify the trigger, explain the mechanism, perform the decision under time pressure and reject an attractive answer that exceeds the evidence.
- Start with the exact mistake and the evidence boundary.
- Build the mechanism before memorising a shortcut.
- Study worked contrasts and name why the near-miss fails.
- Repair misconceptions with one targeted check each.
- Attempt graded independent practice before reading the explanations.
- Return after a delay and prove that the method transfers.
- Use the parent and tutor notes after the learner has attempted the work.
The problem beneath the mistake
A positive whole number has exactly fifteen factors when its prime-exponent choice product equals 15. The target counts combinations of exponent choices.
For n=pᵃqᵇ…, the number of positive factors is (a+1)(b+1)… once n is expressed with distinct prime bases.
The multiplicative partitions of 15 using integers greater than one are 15 and 5×3. Reversing 5 and 3 does not create a new structural family.
The single choice factor 15 gives a+1=15, so a=14. This creates n=p¹⁴ for a prime p.
The two-factor pattern 5×3 gives exponents 4 and 2. This creates n=p⁴q², with p and q distinct primes.
There is no three-prime structure because 15 cannot be written as a product of three integers greater than one. Every additional prime base would add another choice factor.
The prime bases must be distinct. If p=q, then p⁴q² combines to p⁶, which has seven factors rather than fifteen.
The roles can be renamed: p²q⁴ belongs to the same exponent-pattern family after exchanging the names p and q.
The smallest mixed example is 2⁴×3²=144. Its choice product is 5×3=15, so 144 has exactly fifteen positive factors.
A prime fourteenth power also qualifies, but the smallest one, 2¹⁴, is much larger than 144. Structure, not simple size order, determines the count.
Every qualifying number is a perfect square because all exponents are even. However, most perfect squares do not have fifteen factors.
The reliable routine is factor completely, form the choice product, partition 15 as 15 or 5×3, subtract one, enforce distinct primes and verify both directions.
Core decision: A positive whole number has exactly fifteen positive factors if and only if it has form p¹⁴ or p⁴q², where p and q are distinct primes. The forms come from the complete multiplicative partitions 15 and 5×3.
Exact divisor classification converts a number into independent prime-exponent choices. The multiplicative structure of the target count determines how many prime bases can occur and what their exponents must be.
Mechanism: why the rule works
1. Factor canonically
Express the number with distinct prime bases.
Composite bases and repeated primes must be resolved before counting.
Diagnostic check. Are all bases prime and combined?
2. Count choices
Exponent a permits divisor exponents 0 through a, giving a+1 choices.
Independent choices multiply.
Diagnostic check. What is the full choice product?
3. Partition fifteen
Use 15 and 5×3.
These are the only unordered products above one.
Diagnostic check. Have both partitions been included?
4. Translate exponents
Subtract one from every choice factor.
Fifteen gives 14; five and three give 4 and 2.
Diagnostic check. Was one subtracted from each factor?
5. Require distinct bases
The mixed structure needs different primes.
Equal bases combine and destroy independence.
Diagnostic check. Are p and q distinct?
6. Verify sufficiency
Compute (14+1) or (4+1)(2+1).
Both return fifteen.
Diagnostic check. Does the form reproduce the target?
7. Prove completeness
No other multiplicative partition of fifteen exists.
Therefore no other exponent pattern exists.
Diagnostic check. What partition would a proposed extra form require?
Worked examples: make the decision visible
Pause before each answer. State the controlling evidence in one line, then complete the operation. The explanations deliberately contrast a valid route with a plausible but invalid one, because transfer grows when the learner can say why a tempting route does not apply.
Worked example 1: Mixed example
Task. 144=2⁴×3².
Reasoning. The choice product is 5×3.
Answer. 144 has fifteen positive factors.
Independent check. The primes are distinct.
Contrast-and-transfer pass. Keep the governing principle from “Mixed example”, but change one surface detail in the task: 144=2⁴×3². Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—144 has fifteen positive factors.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 2: Prime power
Task. 2¹⁴ has exponent choices 0 through 14.
Reasoning. There are fifteen choices.
Answer. It qualifies.
Independent check. One base matches the 15 pattern.
Contrast-and-transfer pass. Keep the governing principle from “Prime power”, but change one surface detail in the task: 2¹⁴ has exponent choices 0 through 14. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—It qualifies.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 3: Renamed roles
Task. 2²×3⁴ has exponents 2 and 4.
Reasoning. Rename p=3 and q=2.
Answer. It belongs to p⁴q².
Independent check. It is not a third family.
Contrast-and-transfer pass. Keep the governing principle from “Renamed roles”, but change one surface detail in the task: 2²×3⁴ has exponents 2 and 4. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—It belongs to p⁴q².—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 4: Wrong exponent
Task. p¹⁵ has sixteen factors.
Reasoning. Choices run from 0 through 15.
Answer. It does not qualify.
Independent check. Subtract one from the target choice count.
Contrast-and-transfer pass. Keep the governing principle from “Wrong exponent”, but change one surface detail in the task: p¹⁵ has sixteen factors. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—It does not qualify.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 5: Equal primes
Task. p⁴×p²=p⁶.
Reasoning. Like bases combine.
Answer. p⁶ has seven factors.
Independent check. Distinctness is essential.
Contrast-and-transfer pass. Keep the governing principle from “Equal primes”, but change one surface detail in the task: p⁴×p²=p⁶. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—p⁶ has seven factors.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 6: Three bases
Task. pqr has 2×2×2=8 factors.
Reasoning. Three exponent-one primes do not reach fifteen.
Answer. It does not qualify.
Independent check. Fifteen has no three-factor partition above one.
Contrast-and-transfer pass. Keep the governing principle from “Three bases”, but change one surface detail in the task: pqr has 2×2×2=8 factors. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—It does not qualify.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 7: Candidate 400
Task. 400=2⁴×5².
Reasoning. The choice product is 5×3.
Answer. 400 qualifies.
Independent check. The same mixed pattern appears.
Contrast-and-transfer pass. Keep the governing principle from “Candidate 400”, but change one surface detail in the task: 400=2⁴×5². Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—400 qualifies.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 8: Candidate 225
Task. 225=3²×5².
Reasoning. The choice product is 3×3=9.
Answer. It does not qualify.
Independent check. Being a square is not sufficient.
Contrast-and-transfer pass. Keep the governing principle from “Candidate 225”, but change one surface detail in the task: 225=3²×5². Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—It does not qualify.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 9: Composite base
Task. 4¹⁴=2²⁸.
Reasoning. Complete factorisation gives twenty-nine choices.
Answer. It does not qualify.
Independent check. The p in p¹⁴ must be prime.
Contrast-and-transfer pass. Keep the governing principle from “Composite base”, but change one surface detail in the task: 4¹⁴=2²⁸. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—It does not qualify.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 10: Smallest reasoning
Task. Assign exponent 4 to the smallest prime 2 and exponent 2 to 3.
Reasoning. 2⁴×3²=144.
Answer. This is smaller than swapping exponents.
Independent check. Compare structures, not just bases.
Contrast-and-transfer pass. Keep the governing principle from “Smallest reasoning”, but change one surface detail in the task: Assign exponent 4 to the smallest prime 2 and exponent 2 to 3. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—This is smaller than swapping exponents.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 11: Square consequence
Task. Are all qualifying numbers squares?
Reasoning. Yes, because exponents 14 or 4 and 2 are even.
Answer. The converse is false.
Independent check. Odd factor count and square structure align.
Contrast-and-transfer pass. Keep the governing principle from “Square consequence”, but change one surface detail in the task: Are all qualifying numbers squares? Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—The converse is false.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 12: Completeness
Task. Could exponents 6 and 1 work?
Reasoning. They give 7×2=14 factors.
Answer. No.
Independent check. Match the target product exactly.
Contrast-and-transfer pass. Keep the governing principle from “Completeness”, but change one surface detail in the task: Could exponents 6 and 1 work? Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—No.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
The method lab: change one condition at a time
Build three controlled cases around this problem: the learner writes only p¹⁴ or guesses exponent pairs that add to fifteen, instead of translating the multiplicative partitions 15 and 5×3 into divisor choices. Include one valid example, one near-miss and one boundary case. Change only one decisive condition at a time and name the evidence that changes the decision.
Turn the central claim into a decision card: A positive whole number has exactly fifteen positive factors if and only if it has form p¹⁴ or p⁴q², where p and q are distinct primes. The forms come from the complete multiplicative partitions 15 and 5×3. Put the trigger on the front. On the back, write the mechanism, one worked example, one tempting wrong answer and the limit.
Make an evidence ladder with four rungs: given information or observation, interpreted relationship, conclusion and independent check. Rewrite any rung that merely repeats another.
Construct two attractive wrong answers. Identify the exact decision where each fails, explain why it looks plausible and write the smallest repair. Change the surface details and repeat.
Teach the idea aloud in ninety seconds without notes. Include the controlling condition, mechanism, example, near-miss and one sentence beginning “This does not mean…”.
After two days, attempt one direct item, one disguised item and one item where the rule should not be used. After one week, rebuild the rule on a blank page and verify it independently.
Common misconceptions and targeted repairs
Misconception 1: Only p¹⁴
What the learner may think. Fifteen is treated as if prime.
Why it fails. 15 also factors as 5×3.
Targeted repair. List multiplicative partitions.
Misconception 2: Use p¹⁵
What the learner may think. The target count becomes the exponent.
Why it fails. Exponent plus one is the choice count.
Targeted repair. Subtract one.
Misconception 3: Add exponents
What the learner may think. Four plus two gives six, not fifteen.
Why it fails. Factor counts multiply exponent choices.
Targeted repair. Use (a+1)(b+1).
Misconception 4: Equal bases allowed
What the learner may think. p and q may be the same.
Why it fails. Like prime powers combine.
Targeted repair. Require distinct primes.
Misconception 5: Order is new
What the learner may think. p⁴q² and p²q⁴ are different families.
Why it fails. Prime names can be exchanged.
Targeted repair. Classify unordered patterns.
Misconception 6: Any square
What the learner may think. All perfect squares have fifteen factors.
Why it fails. Squares have many possible odd factor counts.
Targeted repair. Apply the exact formula.
Misconception 7: Composite p
What the learner may think. Any base works in p¹⁴.
Why it fails. The base must be prime in canonical form.
Targeted repair. Factor the base.
Misconception 8: Examples prove all
What the learner may think. Examples establish completeness.
Why it fails. Only the partition argument rules out other forms.
Targeted repair. Write the structural proof.
Graded independent practice with explained answers
Attempt each item without looking ahead. Write the evidence or relationship before the final response. Early items check recognition; middle items require explanation; later items change the surface form so that copying a worked example is insufficient.
Practice 1: Forms
Question. State both structures.
Explained answer. p¹⁴ or p⁴q² for distinct primes.
Why this earns trust. Both target partitions appear.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “p¹⁴ or p⁴q² for distinct primes.” from memory, but to recreate a justified answer from the new evidence.
Practice 2: Partitions
Question. How is fifteen split?
Explained answer. 15 and 5×3.
Why this earns trust. Order does not matter.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “15 and 5×3.” from memory, but to recreate a justified answer from the new evidence.
Practice 3: Single exponent
Question. What exponent comes from fifteen choices?
Explained answer. 14.
Why this earns trust. Subtract one.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “14.” from memory, but to recreate a justified answer from the new evidence.
Practice 4: Mixed exponents
Question. What exponents come from five and three choices?
Explained answer. 4 and 2.
Why this earns trust. Subtract one from each.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “4 and 2.” from memory, but to recreate a justified answer from the new evidence.
Practice 5: Check 144
Question. Does 144 qualify?
Explained answer. Yes, 144=2⁴×3².
Why this earns trust. Its choice product is fifteen.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “Yes, 144=2⁴×3².” from memory, but to recreate a justified answer from the new evidence.
Practice 6: Check p¹⁵
Question. Does p¹⁵ qualify?
Explained answer. No, it has sixteen factors.
Why this earns trust. Count zero through fifteen.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “No, it has sixteen factors.” from memory, but to recreate a justified answer from the new evidence.
Practice 7: Distinctness
Question. Why must p differ from q?
Explained answer. Otherwise powers combine.
Why this earns trust. The count changes.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “Otherwise powers combine.” from memory, but to recreate a justified answer from the new evidence.
Practice 8: Three primes
Question. Can three distinct primes occur?
Explained answer. No.
Why this earns trust. Fifteen has no three-factor partition above one.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “No.” from memory, but to recreate a justified answer from the new evidence.
Practice 9: Square
Question. Is every qualifying number a square?
Explained answer. Yes.
Why this earns trust. All allowed exponents are even.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “Yes.” from memory, but to recreate a justified answer from the new evidence.
Practice 10: Converse
Question. Does every square qualify?
Explained answer. No.
Why this earns trust. Exact exponents still matter.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “No.” from memory, but to recreate a justified answer from the new evidence.
Practice 11: Rename
Question. Is p²q⁴ a new family?
Explained answer. No.
Why this earns trust. Exchange the prime names.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “No.” from memory, but to recreate a justified answer from the new evidence.
Practice 12: Formula
Question. State the divisor formula.
Explained answer. Multiply one more than each prime exponent.
Why this earns trust. Use canonical prime factors.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “Multiply one more than each prime exponent.” from memory, but to recreate a justified answer from the new evidence.
Practice 13: Completeness
Question. Why are there only two forms?
Explained answer. Fifteen has only two multiplicative partitions above one.
Why this earns trust. The exponent patterns are forced.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “Fifteen has only two multiplicative partitions above one.” from memory, but to recreate a justified answer from the new evidence.
Practice 14: Routine
Question. State the method.
Explained answer. Factor, count choices, partition 15, subtract one, check distinct primes and verify.
Why this earns trust. Every gate is explicit.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “Factor, count choices, partition 15, subtract one, check distinct primes and verify.” from memory, but to recreate a justified answer from the new evidence.
A four-session mastery cycle
Session 1: construct. Read the mechanism once, cover it and write a three-line model containing the trigger, the operation and the check. Add one example and one near-miss. Compare only after the model is complete. If the operation is right but the trigger is missing, the knowledge is unsafe because it may be applied in the wrong place.
Session 2: discriminate. Mix six items: two that clearly require this method, two that tempt the method but violate a condition, and two that require another skill. For every item, write “use” or “do not use” before answering. Accuracy in method selection matters more than speed at this stage.
Session 3: explain. Give a complete answer to an imagined classmate. Include the smallest sufficient evidence, the mechanism and the independent check. Remove any sentence that merely repeats the question or adds an unsupported claim. A concise causal explanation is stronger than a longer collection of familiar words.
Session 4: perform. Attempt a timed mixed set without notes. Mark uncertain decisions, but continue. During review, classify each error as trigger, evidence, operation, scope, unit or check. Reattempt one changed version after a delay. Mastery is shown when the learner chooses and adapts the method without needing the original example beside it.
Delayed transfer: prove that the learning lasts
Transfer test 1: Changed context
Apply the idea in a fresh context after two days.
Success criterion. The trigger, mechanism and limit are all stated.
Transfer test 2: Near-miss audit
Explain why a tempting example fails.
Success criterion. The exact failed condition is named.
Transfer test 3: Independent check
Verify the conclusion by a second route.
Success criterion. The check does not repeat the same possible mistake.
Transfer test 4: Blank recall
Rebuild the idea after one week.
Success criterion. Definition, example, near-miss and boundary appear.
Examination control
1. Factor completely. Use distinct prime bases.
2. Write the choice product. Use exponent plus one.
3. List 15 and 5×3. Do not omit the mixed form.
4. Check distinct primes. Prevent combining powers.
5. Verify both forms. Return exactly fifteen.
Under time pressure, compress the method to choice product 15 → 15 or 5×3 → exponents 14 or 4,2 → distinct-prime check. The compressed routine is useful only after the full reasoning is understood. If it produces uncertainty, expand the step instead of guessing.
A compact error log
Label the failed decision rather than writing “careless”. A precise error label makes the next practice item useful.
Reattempt one changed version after a delay. Mastery requires method selection, not recognition of the original wording.
Notes for parents and tutors
Ask for controlling evidence before accepting the answer.
Let the learner attempt a near-miss before explaining again.
Praise precise limits and checks, not speed alone.
Use fresh contexts so the decision transfers.
Frequently asked questions
What forms have fifteen factors?
p¹⁴ and p⁴q² for distinct primes.
Why two forms?
Because 15=15 and 15=5×3.
Why exponent fourteen?
There are choices zero through fourteen.
Why exponents four and two?
Five and three choices become four and two.
May p equal q?
No.
What is a small example?
144=2⁴×3².
Are all qualifying numbers squares?
Yes.
Does every square qualify?
No.
Next routes
Return to the complete volume directory to locate the series, use the original PSLE Learning Guide for the foundation route, and continue through PSLE Mathematics for verified subject guidance. The next useful task is a mixed set in which this idea is sometimes relevant and sometimes irrelevant; that is where the learner must choose a method rather than imitate one.