Before you begin
What you need: Whole-number addition, multiplication and division.
- Distinguish integers, primes and composite numbers
- Find factors and use divisibility tests
- Express division with a quotient and remainder
Understand the idea
Know the number families
Whole numbers are 0, 1, 2, …; integers also include negatives. Rational numbers can be written p/q with integer p and nonzero integer q. Irrational numbers, such as √2, cannot be expressed this way.
Factors come in pairs
A factor divides a number exactly. For 24, the pairs are 1 × 24, 2 × 12, 3 × 8 and 4 × 6. A prime has exactly two positive factors; a composite number has more than two. The number 1 is neither prime nor composite.
Remainders tell what is left
When dividing a nonnegative integer by a positive integer, the remainder is at least zero and smaller than the divisor. A remainder of zero means exact divisibility.
A quick divisibility toolkit
| Divisor | Test | Example |
|---|---|---|
| 2 | Last digit is even | 348 ends in 8 |
| 3 | Digit sum is divisible by 3 | 348 → 15 |
| 5 | Last digit is 0 or 5 | 735 ends in 5 |
| 9 | Digit sum is divisible by 9 | 729 → 18 |
A method you can reuse
Dividend = divisor × quotient + remainder
For nonnegative integer division: 0 ≤ remainder < divisor.- Identify whether the question asks for factors, multiples or a remainder.
- For prime testing, check prime divisors up to the square root.
- For division, find the greatest multiple no larger than the dividend and subtract.
To test whether an integer n > 1 is prime, it is enough to test prime divisors up to √n: a larger factor would have a smaller partner.
Worked examples
Read the question first. Try a step yourself, then compare your reasoning.
Factor pairs
List all positive factors of 18.
- Find pairs: 1 × 18, 2 × 9, 3 × 6.
- Once the smaller factor exceeds √18, the pairs repeat.
- Collect the factors in increasing order.
Remainder in division
Find the remainder when 257 is divided by 7.
- 7 × 36 = 252.
- 257 − 252 = 5.
- Check: 0 ≤ 5 < 7.
A prime or a composite?
Is 97 prime?
- √97 is between 9 and 10.
- Test primes 2, 3, 5 and 7.
- 97 is divisible by none of them, so it has no factor pair other than 1 × 97.
Check your understanding
Quick questions, with explanations. These are for self-study and do not affect your account score.
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