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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