Factors and multiples
20 mangoes can be packed in boxes in different ways: 4 boxes of 5, 2 boxes of 10 or 1 box of 20. The numbers 1, 2, 4, 5, 10, 20 that make equal packs are the factors of 20.
If a number divides another number exactly, it is a factor of it, and the bigger number is a multiple of it. 3 × 4 = 12, so 3 and 4 are factors of 12 and 12 is a multiple of 3 and of 4. Arranging a number in an array (rows × columns) shows its factors.
Formula: Factor × factor = multiple. 1 and the number itself are always factors. Every number is a multiple of itself.
In the figure: 12 can be arranged as 3 × 4, 2 × 6 or 1 × 12. So its factors are 1, 2, 3, 4, 6 and 12.
Rule 1: Find the factors — Try 1, 2, 3 … in order and keep those that divide exactly. Stop when the pairs start to repeat.
Factors and multiples
In the app you can move this figure with your finger.
Open in the app- Easy: 18: 1 × 18, 2 × 9, 3 × 6, so 1, 2, 3, 6, 9, 18.
- Medium: 20: 1, 2, 4, 5, 10, 20.
- Exam-type: 30: 1, 2, 3, 5, 6, 10, 15, 30.
Rule 2: Find the multiples — Multiply the number by 1, 2, 3, 4 … The list never ends and each is bigger than or equal to the number.
- Easy: Multiples of 7: 7, 14, 21, 28, 35 …
- Medium: Multiples of 9: 9, 18, 27, 36, 45 …
- Exam-type: Is 54 a multiple of 6? 6 × 9 = 54, yes.
Factors are small or equal to the number; there are only a few. Multiples are big or equal; there are infinitely many.
Missing the pairs, e.g. forgetting 9 and 18 for the number 18, or calling 0 a factor.
Write factors in pairs: 1 × n, 2 × …, 3 × … Memory clue: "Factors few and small, multiples many and tall."
Spot the difference
| Number | Factors |
|---|---|
| 12 | 1, 2, 3, 4, 6, 12 |
| 18 | 1, 2, 3, 6, 9, 18 |
| 20 | 1, 2, 4, 5, 10, 20 |
| 30 | 1, 2, 3, 5, 6, 10, 15, 30 |
Practice questions
- 1List the factors of 16.
- 2List the factors of 21.
- 3Write the first five multiples of 8.
- 4Is 45 a multiple of 9?
- 5How many factors does 28 have?
- 6Draw two different arrays for 15 and name the factors.
- 7Smallest multiple of 7 greater than 60?
- 8Find a number less than 50 that has exactly 3 factors? (Hint: 3 factors means a square of a prime.)
Show answers (try them yourself first)
- 11, 2, 4, 8, 16
- 21, 3, 7, 21
- 38, 16, 24, 32, 40
- 4Yes, 9 × 5 = 45
- 56 (1, 2, 4, 7, 14, 28)
- 63 × 5 and 1 × 15; factors 1, 3, 5, 15
- 763
- 84 (1, 2, 4), 9, 25 or 49
Common multiples
A rabbit jumps 4 at a time and a frog jumps 3 at a time. Both start at 0. The first spot where they land together is 12; they meet again at 24, 36 and so on.
A number that is a multiple of two or more numbers is their common multiple. The smallest common multiple (after 0) is the first one; all others are multiples of it.
Formula: Common multiples of a and b: numbers in both lists of multiples. The first one is at most a × b.
In the figure: Rabbit jumps 4, 8, 12 and frog jumps 3, 6, 9, 12: both touch 12. Common multiples of 4 and 6 are 12, 24, 36 …
Rule 1: List and match — Write the multiples of both numbers and pick the numbers that appear in both.
Common multiples
In the app you can move this figure with your finger.
Open in the app- Easy: Multiples of 2: 2, 4, 6, 8, 10, 12; of 5: 5, 10, 15: first common 10.
- Medium: 6 and 8: 24 is the first common multiple (6 × 4 = 8 × 3).
- Exam-type: 3 and 9: the first common multiple is 9 itself.
Rule 2: Meeting together — Things that repeat (jumps, bells, hunting days) meet again at the common multiples.
- Easy: Bells every 6 and 8 minutes ring together after 24 minutes.
- Medium: Tiger every 3 days, panther every 5: together on day 15, 30 …
- Exam-type: Buses every 4 and 10 minutes leave together every 20 minutes.
There are infinitely many common multiples; they are all multiples of the first one.
Taking the product a × b as always the first common multiple (it is not for 4 and 6), or starting the list from 0.
If the bigger number is a multiple of the smaller, it is the answer. Memory clue: "Both lists, same number: meet there."
Spot the difference
| Pair | First common multiple |
|---|---|
| 3, 4 | 12 |
| 4, 6 | 12 |
| 6, 8 | 24 |
| 3, 9 | 9 |
Practice questions
- 1First common multiple of 2 and 5?
- 2First common multiple of 3 and 4?
- 3Write 3 common multiples of 2 and 3.
- 4First common multiple of 6 and 9?
- 5A bell rings every 4 minutes and another every 6 minutes. They ring together at 0. When do they ring together next?
- 6A deer jumps 5 and a goat jumps 4 from 0. Where do they first land together?
- 7Two friends exercise every 3rd and every 7th day starting together. On which days are they together up to day 50?
- 8Is 36 a common multiple of 4 and 6? Is it the first one?
Show answers (try them yourself first)
- 110
- 212
- 36, 12, 18
- 418
- 5After 12 minutes
- 620
- 7Days 21 and 42
- 8Yes, but not the first; the first is 12
Common factors
Mowgli can jump 2 steps at a time and land on both 24 and 36. He can also jump by 3, 4, 6 or 12. These jump sizes divide both numbers: they are common factors.
A common factor of two or more numbers divides each of them exactly. 1 is a common factor of every pair. The biggest of the common factors is the greatest common factor.
Formula: Common factors = factors that appear in both lists. Two consecutive numbers have only 1 as a common factor.
In the figure: Factors of 24 and 36 share 1, 2, 3, 4, 6, 12. Jumps of 3 from 0 land on 9, 21, 39: 3 is a common factor of them.
Rule 1: List and compare — Write all the factors of both numbers and circle the ones in both lists.
- Easy: 18 and 27: factors 1, 2, 3, 6, 9, 18 and 1, 3, 9, 27; common 1, 3, 9.
- Medium: 10 and 25: common 1 and 5.
- Exam-type: 7 and 11: only 1.
Rule 2: Quick test — To test one number k, divide both numbers by k: if both divide exactly, k is a common factor.
- Easy: Is 5 a common factor of 30 and 45? Yes.
- Medium: Is 7 a common factor of 36 and 60? No.
- Exam-type: Is 4 a common factor of 18 and 30? No: 4 does not divide 18.
The greatest common factor is never bigger than the smaller of the two numbers.
Listing multiples instead of factors, or forgetting that 1 is always a common factor.
Factor the smaller number first, then test each factor on the bigger one. Memory clue: "Common factor: divides both."
Spot the difference
| Pair | Common factors |
|---|---|
| 12, 18 | 1, 2, 3, 6 |
| 20, 30 | 1, 2, 5, 10 |
| 14, 21 | 1, 7 |
| 7, 11 | 1 |
Practice questions
- 1Common factors of 12 and 18?
- 2Common factors of 20 and 30?
- 3Is 3 a common factor of 27 and 40?
- 4Common factors of 13 and 14?
- 5Greatest common factor of 16 and 24?
- 6A frog jumps equal steps from 0 and lands on both 28 and 42. Give three possible jump sizes.
- 7Can a jump size of 5 reach both 25 and 60? Can 10?
- 8Cakes: 24 plain and 36 chocolate are packed in identical boxes with no leftover, each box having only one kind. What is the largest number of cakes in a box?
Show answers (try them yourself first)
- 11, 2, 3, 6
- 21, 2, 5, 10
- 3No: 3 does not divide 40
- 4Only 1
- 58
- 6Any of 1, 2, 7, 14
- 75: yes; 10: no (25 is not a multiple of 10)
- 812 (the greatest common factor)
Divisibility by 2, 5 and 10
At a shop, items costing ₹40 can be paid with 4 notes of ₹10, 8 notes of ₹5 or 20 coins of ₹2. A quick look at the last digit tells you which notes can pay exactly.
A number is divisible by another if it leaves no remainder. Quick tests by the last digit: divisible by 2 if it ends in 0, 2, 4, 6 or 8; by 5 if it ends in 0 or 5; by 10 if it ends in 0.
Formula: Ends in 0 → divisible by 2, 5 and 10. Ends in 5 → divisible by 5 only (of these). Ends in 2, 4, 6, 8 → divisible by 2 only.
In the figure: Last digit tells all: 0, 2, 4, 6, 8 for 2; 0, 5 for 5; only 0 for 10. 80 ends in 0, so it is divisible by all three.
Rule 1: Sort by last digit — Look only at the ones digit. 0 → all three; 5 → only 5; even digit other than 0 → only 2.
- Easy: 60, 130, 480: divisible by 2, 5 and 10.
- Medium: 35, 85, 145: divisible by 5 only.
- Exam-type: 62, 84, 98: divisible by 2 only.
Rule 2: Why the rule works — Tens, hundreds … are all multiples of 10, so only the ones digit decides divisibility by 2, 5 and 10.
- Easy: 74 = 70 + 4: 70 is a multiple of 10, 4 is even: 74 is divisible by 2.
- Medium: 85 = 80 + 5: divisible by 5, not by 10.
- Exam-type: 100, 110, 120 … are all multiples of 10.
A number divisible by 10 is divisible by both 2 and 5 (10 = 2 × 5).
Thinking every number ending in 5 is divisible by 10, or an odd number by 2.
Cover everything except the last digit. Memory clue: "Last digit decides 2, 5, 10."
Spot the difference
| Divisor | Last digit |
|---|---|
| 2 | 0, 2, 4, 6, 8 |
| 5 | 0, 5 |
| 10 | 0 |
Practice questions
- 1Is 345 divisible by 5?
- 2Is 678 divisible by 2? By 5?
- 3Is 450 divisible by 10?
- 4Sort 90, 22, 75, 40, 55, 38 into: only 2 / only 5 / 2, 5 and 10.
- 5Write a 3-digit number divisible by 2, 5 and 10.
- 6Smallest 3-digit number divisible by 5?
- 7A number is divisible by 5 but not by 10 and lies between 60 and 80. Which?
- 8Is the sum 35 + 45 divisible by 10? Is 35 × 2?
Show answers (try them yourself first)
- 1Yes (ends in 5)
- 2By 2: yes; by 5: no
- 3Yes (ends in 0)
- 4Only 2: 22, 38; only 5: 75, 55; all: 90, 40
- 5For example 230 (any number ending in 0)
- 6100
- 765 or 75
- 880: yes; 70: yes
Prime and composite numbers
13 sweets can be arranged only as 1 row of 13 or 13 rows of 1: no neat rectangle. 12 sweets can make 2 × 6, 3 × 4 and more. This tells primes from other numbers.
A prime number has exactly two factors: 1 and itself (13, 37). A number with more than two factors is composite (12, 21). 1 has only one factor, so it is neither prime nor composite.
Formula: Prime: 2 factors. Composite: more than 2 factors. 1: one factor. 2 is the only even prime.
In the figure: Primes up to 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 (green). 1 is neither; the rest are composite.
Rule 1: Test a number — Try dividing by 2, 3, 5, 7 …. If any divides it exactly (and is smaller than it), the number is composite.
Prime and composite numbers
In the app you can move this figure with your finger.
Open in the app- Easy: 21 = 3 × 7, composite.
- Medium: 29: not divisible by 2, 3, 5: prime.
- Exam-type: 51 = 3 × 17, composite (looks prime but is not).
Rule 2: Special numbers — 1 is neither prime nor composite. 2 is the smallest prime and the only even prime. All other primes are odd.
- Easy: Primes between 30 and 50: 31, 37, 41, 43, 47.
- Medium: Even numbers above 2 are composite: 4, 6, 8 …
- Exam-type: Numbers ending in 5 above 5 are composite: 15, 25, 35 …
Every number above 1 is either prime or composite.
Calling 1 prime, 9 prime (it is 3 × 3), or thinking 2 is composite.
Check divisibility by 2, 3, 5, 7 only; that is enough for numbers up to 100. Memory clue: "Prime: only 1 and me."
Spot the difference
| Type | Factors | Example |
|---|---|---|
| Prime | 2 | 13 |
| Composite | > 2 | 21 |
| Neither | 1 | 1 |
Practice questions
- 1Is 17 prime?
- 2Is 27 prime? Give its factors.
- 3List all primes less than 20.
- 4Which is the only even prime?
- 5Why is 1 not prime?
- 6Find the primes between 40 and 44.
- 7Write 24 as a product of two numbers in 3 different ways.
- 8Two primes add up to 20. Find them.
Show answers (try them yourself first)
- 1Yes, factors 1 and 17
- 2No; 1, 3, 9, 27
- 32, 3, 5, 7, 11, 13, 17, 19
- 42
- 5It has only one factor
- 641 and 43
- 71 × 24, 2 × 12, 3 × 8, 4 × 6
- 83 + 17 or 7 + 13
Jump and pattern problems
Animals race along a trail: each takes equal jumps. Who reaches the food? Where do they meet? These puzzles use factors and multiples together.
If a jumper takes jumps of size k from 0, it lands exactly on the multiples of k. A spot n is reached if k is a factor of n; the number of jumps is n ÷ k. Two jumpers meet at common multiples.
Formula: Jumps needed = n ÷ k. Reach n if k is a factor of n. Meet at common multiples of the jump sizes.
In the figure: Jumps of 3 touch 3, 6, 9, 12 …: 9, 21 and 39 are reached; 25 is not.
Rule 1: Who reaches the food? — The food at n is reached by every jumper whose jump size divides n. The bigger jump needs fewer jumps.
Jump and pattern problems
In the app you can move this figure with your finger.
Open in the app- Easy: Food at 36: jumps of 4 (9 jumps) and 6 (6 jumps) both reach it; 6 gets there first.
- Medium: Food at 30: jumps of 4 never land on 30, so that animal misses it.
- Exam-type: Food at 42: jumps of 7 (6 jumps), 6 (7 jumps), 3, 2, 14, 21 also work.
Rule 2: Avoid or meet — To meet a friend at n but avoid another at m, choose a factor of n that is not a factor of m.
- Easy: Meet at 30, avoid 25: jump 3 (10 jumps) works; 5 would land on 25.
- Medium: Meet at 21 and 35: 7 is a common factor (3 and 5 jumps).
- Exam-type: Meet at 24 and 36 only with jumps of 1, 2, 3, 4, 6 or 12.
Always check by dividing: n ÷ k must come out with no remainder.
Counting jumps from 1 instead of 0, or picking a jump that lands on the friend to be avoided.
Divide the target by the jump size. Memory clue: "Jump divides target: reach it."
Spot the difference
| Question | Tool |
|---|---|
| Does k reach n? | k is a factor of n |
| When do two meet? | common multiple |
| One jump for both targets? | common factor |
Practice questions
- 1How many jumps of 7 reach 56?
- 2Can jumps of 6 reach 45?
- 3Food at 48. Do jumps of 4 and 6 both reach it? Who is quicker?
- 4Which jump sizes (other than 1) reach both 12 and 18?
- 5Zara jumps 5 and Yash jumps 3 from 0. Where do they first land together?
- 6Friends live at 8, 12, 15 and 20. Which can be visited with jumps of 4?
- 7Meet at 28, avoid 21. Name a jump size that works.
- 8A frog jumps by 9 and a toad by 6, both from 0. On which numbers do they land on the same stone? Give the first three.
Show answers (try them yourself first)
- 18
- 2No, 45 is not a multiple of 6
- 3Yes; 6 needs 8 jumps, 4 needs 12
- 42, 3, 6
- 515
- 68, 12, 20
- 72 or 4 (7 and 1 would land on 21)
- 818, 36, 54