Mental Ability · Lesson 3

Factors, Multiples & Divisibility

Numbers have families! Discover factors and multiples, spot primes in a flash, grow factor trees, find HCF and LCM, and use divisibility rules to crack missing-digit puzzles.

~45 min read8 quick checksTest: 20 questions

1Factors

A factor of a number divides it exactly, leaving no remainder. 4 is a factor of 12 because 12 ÷ 4 = 3 with remainder 0. 5 is not a factor of 12.

Factors come in pairs that multiply to give the number. To find all factors, start from 1 and test 1, 2, 3, … until the pairs start to repeat.

The factor rainbow for 36: 1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6. So 36 has 9 factors. (6 is counted only once.)

✏️ Worked example

Find all the factors of 48.

1 × 48, 2 × 24, 3 × 16, 4 × 12, 6 × 8.
5 and 7 do not divide 48. Next to test is 8 — but 8 is already in a pair, so stop.
Factors: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 → 10 factors.

1 is a factor of every number. Every number is a factor of itself.
The smallest factor is 1, and the greatest factor is the number itself.
A number has a limited (finite) number of factors.
Only special numbers (square numbers like 36, 49, 100) have an odd number of factors, because one pair uses the same number twice.

How many factors does 28 have?

Pairs: 1 × 28, 2 × 14, 4 × 7. Factors: 1, 2, 4, 7, 14, 28 → 6 factors.

2Multiples

A multiple of a number is what you get when you multiply it by 1, 2, 3, 4 … Multiples of 7: 7, 14, 21, 28, 35, … (its times table).

Every number is a multiple of itself (×1) — that is its smallest multiple.
A number has unlimited (infinite) multiples. There is no greatest multiple.
Factors and multiples are partners: 4 is a factor of 12 ⇔ 12 is a multiple of 4.

Remember: Factors are Few and Fit inside; Multiples are Many and go up. Factors ≤ the number; multiples ≥ the number.

✏️ Worked example — counting multiples in a range

How many multiples of 6 lie between 40 and 80?

First multiple after 40: 6 × 7 = 42. Last multiple before 80: 6 × 13 = 78.
They are 6 × 7, 6 × 8, … , 6 × 13. Count from 7 to 13: 13 − 7 + 1 = 7 multiples.
List to check: 42, 48, 54, 60, 66, 72, 78 ✓

⚠️ Olympiad trap: "between"

"Between 40 and 80" does not include 40 and 80 themselves. If a question asks for multiples of 8 between 40 and 80, the answer is 48, 56, 64, 72 — only 4, because 40 and 80 are left out.

3Even and odd, prime and composite

Even numbers are multiples of 2 (end in 0, 2, 4, 6, 8). Odd numbers end in 1, 3, 5, 7, 9.
even + even = even · odd + odd = even · even + odd = odd · odd × odd = odd · even × anything = even.
A prime number has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13.
A composite number has more than two factors. Examples: 4, 6, 8, 9, 10, 12.

The 25 primes up to 100 (green). Cross out 1, then every multiple of 2, 3, 5 and 7 (except those numbers themselves) — whatever is left is prime. This is the Sieve of Eratosthenes.

⚠️ Olympiad trap: the special numbers 1 and 2

1 is neither prime nor composite — it has only one factor. 2 is the smallest prime and the only even prime; every other even number has 2 as an extra factor, so it is composite. Also: the smallest composite number is 4, the smallest odd prime is 3, and the smallest odd composite is 9.

💡 Tip: testing a number under 100

Check if it divides by 2, 3, 5 or 7. If none of them divides it, it is prime. Beware of numbers that look prime: 51 = 3 × 17, 57 = 3 × 19, 87 = 3 × 29, 91 = 7 × 13.

Range	Primes	How many
1 – 10	2, 3, 5, 7	4
11 – 20	11, 13, 17, 19	4
21 – 50	23, 29, 31, 37, 41, 43, 47	7
51 – 100	53, 59, 61, 67, 71, 73, 79, 83, 89, 97	10

Twin primes and co-primes

Twin primes: Two primes with a difference of 2: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73).
Co-primes: Two numbers whose only common factor is 1, like 8 and 15. They do not need to be prime themselves! Any two consecutive numbers (like 20 and 21) are always co-prime, and so are any two different primes.

Which of these is a prime number?

91 = 7 × 13, 87 = 3 × 29 and 57 = 3 × 19 are composite. 83 is not divisible by 2, 3, 5 or 7, so it is prime.

4Factor trees and prime factorisation

Every composite number can be written as a product of primes. This is its prime factorisation. A factor tree helps: split the number into any two factors, then keep splitting until every branch ends in a prime.

Two different trees for 72 give the same primes. The prime factorisation of a number is always the same, however you start.

✏️ Worked example — the division ladder

Find the prime factorisation of 90 by dividing by primes.

90 is even: 90 ÷ 2 = 45.
45: digit sum 9, so divide by 3: 45 ÷ 3 = 15.
15 ÷ 3 = 5, and 5 is prime.
90 = 2 × 3 × 3 × 5.

⚠️ Olympiad trap

In a prime factorisation every factor must be prime. 60 = 4 × 3 × 5 is a correct product, but it is not the prime factorisation because 4 is not prime. The answer is 60 = 2 × 2 × 3 × 5.

Which is the prime factorisation of 60?

2 × 2 × 15
2 × 2 × 3 × 5
4 × 3 × 5
2 × 3 × 10

Only 2 × 2 × 3 × 5 uses primes alone. 15, 4 and 10 are composite.

5Common factors and HCF

A common factor divides both numbers. The biggest one is the HCF — Highest Common Factor (also called GCD or GCF).

Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Common factors sit in the overlap: 1, 2, 3, 6. HCF = 6.

✏️ Worked example — HCF by prime factors

Find the HCF of 36 and 48.

36 = 2 × 2 × 3 × 3. 48 = 2 × 2 × 2 × 2 × 3.
Primes they share: 2, 2 and 3.
HCF = 2 × 2 × 3 = 12.

✏️ Worked example — a real-life HCF

Two ribbons are 36 cm and 48 cm long. They are cut into pieces of equal length with nothing left over. What is the greatest possible length of each piece?

The length must divide both 36 and 48 → a common factor.
"Greatest" → the HCF, which is 12 cm.
Pieces: 36 ÷ 12 = 3 and 48 ÷ 12 = 4, so 7 pieces in all.

✏️ Worked example — HCF with a remainder

What is the greatest number that divides 38 and 62, leaving a remainder of 2 each time?

Take away the remainder first: 38 − 2 = 36 and 62 − 2 = 60. The number divides these exactly.
HCF of 36 and 60: 36 = 2 × 2 × 3 × 3, 60 = 2 × 2 × 3 × 5 → 2 × 2 × 3 = 12.
Answer: 12. Check: 38 = 12 × 3 + 2 ✓ and 62 = 12 × 5 + 2 ✓

What is the HCF of 18 and 30?

Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30. The highest common one is 6. (90 is the LCM.)

6Common multiples and LCM

A common multiple is a multiple of both numbers. The smallest one is the LCM — Lowest (Least) Common Multiple.

Jumps of 4 and jumps of 6 first land on the same number at 12, so LCM(4, 6) = 12. Common multiples keep going: 12, 24, 36 …

💡 Shortcut: list the biggest number's multiples

For LCM(4, 6, 9), write the multiples of 9 — 9, 18, 27, 36 — and stop at the first one that 4 and 6 also divide. LCM = 36.

✏️ Worked example — bells

Three bells ring every 6, 8 and 12 minutes. They ring together at 10:00 a.m. When will they next ring together?

They ring together after a time that is a multiple of 6, 8 and 12 → the LCM.
Multiples of 12: 12, 24 → 24 ÷ 6 = 4 ✓, 24 ÷ 8 = 3 ✓. LCM = 24.
10:00 a.m. + 24 minutes = 10:24 a.m.

HCF or LCM? How to choose

Use HCF when…	Use LCM when…
cutting or sharing into the greatest / largest equal pieces or groups	finding the smallest / least number that several numbers divide
the answer is smaller than (or equal to) the given numbers	things happen again together (bells, lights, buses)

Golden facts

HCF × LCM = product of the two numbers. For 12 and 18: 6 × 36 = 216 = 12 × 18.
The HCF divides the LCM. HCF ≤ the smaller number; LCM ≥ the larger number.
For co-prime numbers, HCF = 1 and LCM = their product: LCM(8, 15) = 120.
If one number divides the other, the smaller is the HCF and the larger is the LCM: HCF(6, 24) = 6, LCM(6, 24) = 24.

✏️ Worked example — find the other number

The HCF of two numbers is 5 and their LCM is 60. One number is 15. Find the other.

Product of the numbers = HCF × LCM = 5 × 60 = 300.
Other number = 300 ÷ 15 = 20.
Check: HCF(15, 20) = 5 ✓, LCM(15, 20) = 60 ✓

What is the LCM of 4 and 10?

Multiples of 10: 10, 20 … 20 ÷ 4 = 5, so LCM = 20. 40 is a common multiple but not the lowest; 2 is the HCF.

7Divisibility rules

These tests tell you if a number divides exactly — without doing the division.

By	Rule	Example
2	Last digit is even (0, 2, 4, 6, 8)	4,578 ✓
3	Sum of digits is divisible by 3	5,142 → 5+1+4+2 = 12 ✓
4	Last two digits form a number divisible by 4 (or are 00)	7,328 → 28 ÷ 4 = 7 ✓
5	Last digit is 0 or 5	9,845 ✓
6	Divisible by both 2 and 3	438 → even, 4+3+8 = 15 ✓
8	Last three digits form a number divisible by 8 (or are 000)	7,256 → 256 ÷ 8 = 32 ✓
9	Sum of digits is divisible by 9	7,821 → 7+8+2+1 = 18 ✓
10	Last digit is 0	8,490 ✓
11	(Sum of digits in odd places) − (sum in even places), counting from the right, is 0 or a multiple of 11	91,806 → (6+8+9) − (0+1) = 22 ✓

Remember: 2, 5, 10 look at the last digit; 4 looks at the last two; 8 looks at the last three; 3 and 9 add up all the digits.

Combining rules

If a number is divisible by two co-prime numbers, it is divisible by their product: by 2 and 3 → by 6; by 3 and 4 → by 12; by 3 and 5 → by 15.

⚠️ Olympiad trap: only for co-primes

12 is divisible by 4 and by 6, but not by 24! 4 and 6 share the factor 2, so you cannot just multiply them. The safe answer is their LCM: a number divisible by 4 and 6 must be divisible by 12. Also, every number divisible by 9 is divisible by 3, but not the other way round (12 is divisible by 3, not by 9).

Which number is divisible by 9?

4,572
5,431
6,302
7,215

Digit sums: 4+5+7+2 = 18 (divisible by 9 ✓), 5+4+3+1 = 13, 6+3+0+2 = 11, 7+2+1+5 = 15. Only 4,572 works.

8Missing-digit puzzles

Olympiads hide a digit and ask you to find it. Use the right divisibility rule and try the digits 0 to 9.

✏️ Worked example — by 9

Find the digit * so that 5*72 is divisible by 9.

Sum of known digits: 5 + 7 + 2 = 14.
The next multiple of 9 after 14 is 18, so * = 18 − 14 = 4. (27 would need * = 13 — impossible.)
Check: 5,472 ÷ 9 = 608 ✓

✏️ Worked example — by 3 (several answers)

3*46 is divisible by 3. What are the smallest and greatest possible digits?

Known digits: 3 + 4 + 6 = 13.
Sum must be 15, 18 or 21 → * = 2, 5 or 8.
Smallest = 2, greatest = 8. (For 3 and 9, answers repeat every 3 or 9 steps.)

✏️ Worked example — by 8

What is the smallest digit * that makes 13*6 divisible by 8?

Only the last three digits matter: 3*6.
Try: 306 ÷ 8, 316 ÷ 8, 326 ÷ 8 leave remainders; 336 ÷ 8 = 42 ✓
Smallest digit = 3.

✏️ Worked example — by 11

Find * so that 8*62 is divisible by 11.

Places from the right: 2 (1st), 6 (2nd), * (3rd), 8 (4th).
Odd places: 2 + *. Even places: 6 + 8 = 14.
14 − (2 + *) must be 0 or 11 → 12 − * = 0 or 11 → * = 12 (impossible) or * = 1.
Check: 8,162 ÷ 11 = 742 ✓

✏️ Worked example — by 6

Which digits can go in 2*4 to make it divisible by 6?

By 2: it ends in 4, so it is even ✓ for any digit.
By 3: 2 + * + 4 = 6 + * must be a multiple of 3 → * = 0, 3, 6 or 9.
Numbers: 204, 234, 264, 294 — all divisible by 6.

What digit makes 7*43 divisible by 11?

From the right: odd places 3 + *, even places 4 + 7 = 11. 11 − (3 + *) = 8 − * must be 0, so * = 8. Check: 7,843 ÷ 11 = 713.

A number is divisible by both 4 and 6. It must also be divisible by:

4 and 6 are not co-prime, so we use their LCM, 12. The number 12 itself is divisible by 4 and 6 but not by 24 or 36.

9Watch and explore

10Quick recap

A factor divides a number exactly. Factors come in pairs; there are only a few of them.
Multiples come from times tables; there are infinitely many. Smallest multiple = the number itself.
Prime: exactly 2 factors. Composite: more than 2 factors.
1 is neither prime nor composite. 2 is the smallest and only even prime.
There are 25 primes up to 100 (15 of them up to 50).
Twin primes differ by 2. Co-primes have HCF 1 (they need not be prime).
Prime factorisation uses only primes: 72 = 2 × 2 × 2 × 3 × 3.
HCF = biggest common factor → "greatest equal pieces". LCM = smallest common multiple → "together again".
HCF × LCM = product of the two numbers.
2, 5, 10: last digit. 4: last two digits. 8: last three digits.
3 and 9: digit sum. 6: both 2 and 3. 11: alternate-place difference is 0 or a multiple of 11.
Combine rules only for co-primes (2 × 3 = 6, 3 × 4 = 12); 4 and 6 give 12, not 24.

Flip cards

QuestionHow many factors does 16 have?

Answer5: 1, 2, 4, 8, 16 (4 × 4 counts once).

QuestionSmallest composite number?

Answer4 (1 is neither, 2 and 3 are prime).

QuestionIs 91 prime?

AnswerNo! 91 = 7 × 13.

QuestionPrime factorisation of 100?

Answer2 × 2 × 5 × 5.

QuestionHCF and LCM of 9 and 12?

AnswerHCF 3, LCM 36 (check: 3 × 36 = 108 = 9 × 12).

QuestionIs 5,316 divisible by 8?

AnswerNo — 316 ÷ 8 = 39 remainder 4. (It is divisible by 4.)

QuestionPrimes between 10 and 30?

Answer11, 13, 17, 19, 23, 29 — six primes.

QuestionAre 14 and 15 co-prime?

AnswerYes — consecutive numbers always have HCF 1.