PSHS NCE Number Theory and Remainders Reviewer

TEACHER ABI PSHS NCE · QUANTITATIVE ABILITY

Number Theory and Remainders

Use divisibility, odd-even patterns, and remainders to solve number questions without doing more work than necessary.

Grade 6–7 friendlyFast remainder thinkingRegular practiceQuick Fire + Mastery

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Number Theory and Remainders

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A remainder is what is left after making equal groups

23 ÷ 5 = 4 remainder 3

Five groups of 4 use 20. There are 3 left.

Important: A remainder must always be smaller than the divisor. If you divide by 7, the only possible remainders are 0, 1, 2, 3, 4, 5, or 6.

DO IT FAST

Use MULTIPLE → LEFTOVER → PATTERN

1. MULTIPLE: Find the nearest multiple below the number.
2. LEFTOVER: Subtract to get the remainder.
3. PATTERN: If a number changes, track only the leftover part when possible.
4. CHECK: The remainder must be smaller than the divisor.

WORKED EXAMPLES

Six patterns you should recognize

1. Find a remainder

What is the remainder when 47 is divided by 6?

42 is the nearest multiple of 6 below 47

Remainder = 5

2. Remainder after adding

A number leaves remainder 4 when divided by 7. What remainder does the number plus 5 leave?

4 + 5 = 9 → 9 leaves remainder 2 when divided by 7

2

3. Remainder after multiplying

A number leaves remainder 4 when divided by 7. What remainder does 3 times the number leave?

3 × 4 = 12 → 12 leaves remainder 5 when divided by 7

5

4. Odd and even

Odd + odd = ?

3 + 5 = 8

Even

5. Divisibility shortcut

Is 5,742 divisible by 3?

5 + 7 + 4 + 2 = 18

Yes. 18 is divisible by 3.

6. Possible remainder

Which could be a remainder when dividing by 5?

Possible remainders are only 0, 1, 2, 3, or 4.

4 could be a remainder; 5 or more cannot.

TRY WITH ME

Track only the leftover

A number leaves a remainder of 3 when divided by 5. What remainder will 4 times the number leave when divided by 5?

1. Ignore the full groups of 5. They will still divide evenly.
2. Track only the remainder: 3.
3. Multiply: 4 × 3 = 12.
4. 12 leaves remainder 2 when divided by 5.

Remainder = 2

COMMON TRAPS

Watch out for these

Giving a remainder equal to the divisor.
When dividing by 6, remainder 6 is impossible.

Doing the full multiplication.
If only the remainder matters, track the remainder instead.

Forgetting odd/even patterns.
These can answer some questions instantly.

Using long division when a divisibility rule works.
Check the digits first.

REGULAR PRACTICE

Build the skill

These mix direct remainders, transformed remainders, divisibility, and odd-even reasoning.

QUICK FIRE

Can you spot the remainder fast?

Five short questions. Avoid full calculations when the leftover is enough.

Question 1 of 5

FRESH MASTERY CHECK

Can you do it without hints?

Score 5/5 to verify this competency. If you miss one, a fresh set will be ready.

NEED HELP?

Remainder questions still feel tricky?

Send Teacher Abi the question that confused you and your result. I can show you the fastest leftover method.

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PSHS NCE REVIEW PATH

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Next competency: Algebraic Expressions and Simple Equations

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PSHS NCE Factors, Multiples, and Divisibility Reviewer

TEACHER ABI PSHS NCE · QUANTITATIVE ABILITY

Factors, Multiples, and Divisibility

Find number patterns faster, recognize what divides evenly, and know when to use factors, multiples, GCF, or LCM.

Grade 6–7 friendly Fast divisibility tricks Regular practice Quick Fire + Mastery

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Factors go in. Multiples come out.

4 × 6 = 24

4 and 6 are factors of 24. They divide 24 evenly.

24 is a multiple of 4 and 6.

Think of it this way:
Factors are numbers that can fit evenly into a number.
Multiples are answers you get when you keep multiplying.

Factors of 12
1, 2, 3, 4, 6, 12

Multiples of 12
12, 24, 36, 48, 60...

DO IT FAST

Know what the question is asking

1. “Divides evenly?” Think factor or divisibility.
2. “What comes next?” Think multiples.
3. “Greatest number that divides both?” Think GCF.
4. “First number both can reach?” Think LCM.

Fast divisibility checks:
2 → last digit is even
3 → digit sum is divisible by 3
5 → ends in 0 or 5
9 → digit sum is divisible by 9
10 → ends in 0

WORKED EXAMPLES

Six patterns you should recognize

1. Is it a factor?

Is 6 a factor of 42?

42 ÷ 6 = 7

Yes. There is no remainder.

2. Is it a multiple?

Is 45 a multiple of 9?

9 × 5 = 45

Yes.

3. Divisible by 3

Is 7,245 divisible by 3?

7 + 2 + 4 + 5 = 18

Yes. 18 is divisible by 3.

4. Prime or composite?

Is 29 prime?

The only factors are 1 and 29.

29 is prime.

5. Greatest common factor

Find the GCF of 18 and 24.

Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

GCF = 6.

6. Least common multiple

Find the LCM of 4 and 6.

Multiples of 4: 4, 8, 12, 16...
Multiples of 6: 6, 12, 18...

LCM = 12.

TRY WITH ME

Do we need GCF or LCM?

Two lights blink every 6 seconds and every 8 seconds. If they blink together now, after how many seconds will they blink together again?

1. We need a number that both 6 and 8 can reach.
2. That means we need a common multiple.
3. We want the first one, so we need the least common multiple.

Multiples of 6: 6, 12, 18, 24...
Multiples of 8: 8, 16, 24...
Answer: 24 seconds.

COMMON TRAPS

Watch out for these

Mixing up factors and multiples.
Factors are limited. Multiples keep going.

Using LCM when the problem asks for biggest equal groups.
That usually needs GCF.

Calling 1 a prime number.
Prime numbers have exactly two factors. The number 1 has only one.

Doing long division when a divisibility rule is faster.
Check the digits first.

REGULAR PRACTICE

Build the skill

These mix factors, multiples, divisibility, prime numbers, GCF, and LCM.

QUICK FIRE

Can you spot it fast?

Five short questions. Use the fastest valid method.

Question 1 of 5

FRESH MASTERY CHECK

Can you do it without hints?

Score 5/5 to verify this competency. If you miss one, a fresh set will be ready.

NEED HELP?

Not sure whether a problem needs GCF or LCM?

Send Teacher Abi the question that confused you and your result. I can help you spot the clue faster.

Message Teacher Abi

PSHS NCE REVIEW PATH

Choose your next step

Next competency: Fractions

Previous: Whole Numbers and Place Value PSHS NCE Student Hub

Number Properties UPCAT Reviewer: Divisibility, Factors, Multiples, and Remainders

TEACHER ABI UPCAT MATHEMATICS

Number Properties

Recognize the structure of an integer before doing unnecessary computation.

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Divisibility, Factors, Multiples, and Remainders

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Number properties reveal what an integer must be able to do

Divisibility rules test whether division leaves no remainder. Factors divide a number exactly; multiples are produced by multiplying it by integers.

a = dq + r, where 0 ≤ r < d

Prime factorization makes GCF, LCM, factor counting, perfect squares, and many remainder questions easier to see.

GCF versus LCM

Use GCF when dividing quantities into the greatest number of identical groups. Use LCM when cycles must meet again.

Remainders have a limit

A remainder is always smaller than the positive divisor. A remainder of 7 is impossible if the divisor is 7 or less.

DO IT FAST

Translate the wording before calculating

Greatest identical grouping → GCF. First time cycles coincide → LCM. Leaves r when divided by d → n = dq + r. For divisibility, test the smallest useful rule first.

Why it works

The wording tells you whether you need a common divisor, a common multiple, or a remainder relationship. Naming that structure prevents trial-and-error computation.

WORKED EXAMPLES

Five forms you should recognize

1. Divisibility

Problem: The number 42,57□ is divisible by 5 and 9. Find □.

Last digit: 0 or 5; digit sum: 18+□

Why: Only 0 satisfies both divisibility conditions.

2. Greatest grouping

Problem: Divide 84 red and 126 blue flags into the greatest number of identical sets.

GCF(84,126) = 42

Why: The number of sets must divide both quantities exactly.

3. Meeting cycles

Problem: Lights flash every 18 and 24 seconds. When do they meet again?

LCM(18,24) = 72 seconds

Why: Seventy-two is the first positive time belonging to both schedules.

4. Remainder

Problem: n leaves remainder 5 when divided by 7. Find the remainder of n + 10.

5 + 10 = 15; 15 = 2(7) + 1

Why: Only the leftover part matters, and any total remainder at least 7 is reduced again.

5. Perfect square

Problem: Find the least multiplier that makes 540 a perfect square.

540 = 2²×3³×5 → multiply by 3×5 = 15

Why: Every prime in a perfect square must have an even exponent.

COMMON TRAPS

Check before you commit

• Using LCM when a problem asks for the greatest number of equal groups
• Multiplying cycle lengths instead of finding their least common multiple
• Allowing a remainder equal to or greater than the divisor
• Assuming a number is prime because it is odd
• Forgetting to subtract an announced common remainder before finding a GCF
• Adding prime exponents instead of using the divisor-count formula

FIVE-FORM SKILL CHECK

Do you need the lesson-or just practice?

One original question in each form recommends your next step. It does not yet verify mastery.

CHOOSE YOUR PRACTICE

Work at the level you need.

Foundations

Build the core procedure with immediate explanations.

Core Practice

Use mixed forms with less scaffolding.

UPCAT-Style Transfer

Apply the competency in unfamiliar representations.

FRESH MASTERY CHECK

Ready to verify this competency?

A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.

QUICK ANSWERS

Number Properties FAQ

How do I decide between GCF and LCM?

GCF divides existing quantities into the largest identical units; LCM finds the earliest shared multiple or meeting time.

How far must I test to decide whether a number is prime?

Test prime divisors only up to the square root of the number.

How do I count the factors of a number?

If n = pᵃqᵇ, then it has (a + 1)(b + 1) positive factors because each prime exponent can range from zero to its maximum.

RELATED COMPETENCIES

Continue your mathematics review.

Fractions and DecimalsScientific NotationSequences and PatternsSets and Venn Diagrams

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