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State Board · Class 5 · Mathematics

Lesson 8: Multiples and Factors

5 concepts · 40 practice questions with answers · Free to read

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Multiples

From daily life

A bus leaves a stop every 9 minutes: at 9, 18, 27, 36, 45 minutes and so on. These times are the multiples of 9.

Simple definition

A multiple (गुणज) of a number is what we get when we multiply it by 1, 2, 3, 4 and so on. The smallest multiple of a number is the number itself. The multiples go on for ever.

Two rows of boxes: the first five multiples of 6 and of 8. Both rows contain 24, the lowest common multiple.Multiples of 6 and of 8 on a number strip×6612182430×8816243240Common multiple: 24 (smallest, LCM)

In the figure : The first five multiples of 6 and of 8: both rows contain 24, the smallest common multiple.

Rule 1: List the multiples — Multiply the number by 1, 2, 3, 4, ... in turn.

  • Easy : Multiples of 7: 7, 14, 21, 28, 35.
  • Medium : The first five multiples of 12: 12, 24, 36, 48, 60.
  • Exam-type : A bus leaves every 9 minutes from 6:00. It leaves at 6:09, 6:18, 6:27, 6:36 and 6:45.

Rule 2: Multiples of multiples — If a number is a multiple of 6, it is a multiple of 3 and of 2 as well.

  • Easy : 18 is a multiple of 6, so it is a multiple of 3, since 6 = 3 × 2.
  • Medium : 48 is a multiple of 12, so it is also a multiple of 4 and of 6.
  • Exam-type : 20 is a multiple of 4 and of 5; 100 is a multiple of 20, so 100 is a multiple of 4 and of 5.
Remember

The first multiple of a number is the number itself and it has no last multiple.

Common mistake

Writing 0 as the first multiple or saying that multiples stop at 100. Start from 1 × the number and keep going.

Shortcut

Memory clue: Multiples = times table of the number.

Spot the difference

NumberFirst five multiples
66, 12, 18, 24, 30
88, 16, 24, 32, 40
99, 18, 27, 36, 45

Practice questions

Easy
  1. 1Write the first five multiples of 7.
  2. 2What is the smallest multiple of 15?
  3. 3Is 42 a multiple of 6?
Medium
  1. 4Is 50 a multiple of 8?
  2. 5Write the multiples of 12 that are less than 50.
  3. 6Which numbers appear in both the multiples of 6 and of 8 among the first five?
Challenging
  1. 7A bell rings every 8 minutes. At which minutes does it ring in the first 40 minutes?
  2. 8Aman says 0 is the first multiple of 5. Is that right?
Show answers (try them yourself first)
  1. 17, 14, 21, 28, 35
  2. 215
  3. 3Yes, 6 × 7 = 42.
  4. 4No. 8 × 6 = 48 and 8 × 7 = 56.
  5. 512, 24, 36, 48
  6. 624
  7. 78, 16, 24, 32, 40
  8. 8No, the first multiple of 5 is 5 itself (5 × 1).

Factors

From daily life

Twelve counters can be laid in 1 row of 12, 2 rows of 6, 3 rows of 4, or the other way round. Each row number divides 12 exactly. Those numbers are the factors.

Simple definition

A factor (अवयव) of a number divides it exactly, leaving no remainder. Factors come in pairs that multiply to the number. 1 is a factor of every number and every number is its own largest factor.

12 counters arranged as 1 by 12, 2 by 6 and 3 by 4 rectangles, which shows the factor pairs of 12.1 × 122 × 63 × 4Rows × columns for 12 countersFactors of 12: 1, 2, 3, 4, 6, 12

In the figure : 12 counters can be arranged as 1×12, 2×6 or 3×4, which shows the factor pairs.

Rule 1: Find factor pairs — Try 1, 2, 3, ... and write the pair for each number that divides exactly.

  • Easy : Factors of 12: 1 × 12, 2 × 6, 3 × 4 → 1, 2, 3, 4, 6, 12.
  • Medium : Factors of 18: 1 × 18, 2 × 9, 3 × 6 → 1, 2, 3, 6, 9, 18.
  • Exam-type : Factors of 30: 1 × 30, 2 × 15, 3 × 10, 5 × 6 → 1, 2, 3, 5, 6, 10, 15, 30.
Live figure

Factors

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Rule 2: Factor and multiple — If 7 is a factor of a number, the number is a multiple of 7.

  • Easy : 5 is a factor of 20 because 20 ÷ 5 = 4; so 20 is a multiple of 5.
  • Medium : 6 is a factor of 42, so 42 is a multiple of 6.
  • Exam-type : 9 is not a factor of 40, because 40 ÷ 9 leaves a remainder 4.
Remember

Every factor of a number is not bigger than the number. The factors are finite; multiples are endless.

Common mistake

Missing the pair of a factor: after 3 × 6 = 18 do not forget to write both 3 and 6.

Shortcut

Memory clue: Factors are Few, multiples are many (F-F, M-M).

Spot the difference

NumberFactor pairsAll factors
121×12, 2×6, 3×41, 2, 3, 4, 6, 12
181×18, 2×9, 3×61, 2, 3, 6, 9, 18
241×24, 2×12, 3×8, 4×61, 2, 3, 4, 6, 8, 12, 24

Practice questions

Easy
  1. 1Is 4 a factor of 20?
  2. 2Write all the factors of 10.
  3. 3Write all the factors of 18.
Medium
  1. 4Write all the factors of 24.
  2. 5Which is the smallest factor and which is the biggest factor of 36?
  3. 6Is 9 a factor of 40? Why?
Challenging
  1. 730 children are to sit in equal rows. In how many ways can they sit (rows × children in a row)?
  2. 8A number has factors 1, 2, 4, 8 and 16 only. What is the number?
Show answers (try them yourself first)
  1. 1Yes, 20 ÷ 4 = 5.
  2. 21, 2, 5, 10
  3. 31, 2, 3, 6, 9, 18
  4. 41, 2, 3, 4, 6, 8, 12, 24
  5. 51 and 36
  6. 6No, because 40 ÷ 9 leaves a remainder 4.
  7. 71×30, 2×15, 3×10, 5×6 and the reverse of each: 8 arrangements.
  8. 816

Divisibility tests

From daily life

Before a big feast the cook has to share 126 laddus equally among 3 or 9 children without breaking any. Quick tests tell whether a number divides evenly, without long division.

Simple definition

Divisibility tests (भाज्यतेच्या कसोट्या): by 2 if the last digit is 0, 2, 4, 6 or 8; by 5 if it ends in 0 or 5; by 10 if it ends in 0; by 3 if the digit sum is in the table of 3; by 9 if the digit sum is in the table of 9.

Quick tests: divisible by 2, 5, 10 by the last digit; by 3 and 9 by the sum of digits.÷ 2last digit 0, 2, 4, 6, 8÷ 5last digit 0 or 5÷ 10last digit 0÷ 3digit sum in 3 table: 123 → 6÷ 9digit sum in 9 table: 729 → 18

In the figure : Last-digit tests for 2, 5 and 10, and digit-sum tests for 3 and 9.

Rule 1: Last-digit tests — Look only at the last digit for 2, 5 and 10.

  • Easy : 4,320 ends in 0, so it is divisible by 2, 5 and 10.
  • Medium : 1,235 ends in 5, so it is divisible by 5 but not by 2 or 10.
  • Exam-type : 546 ends in 6, so it is divisible by 2 but not by 5 or 10.

Rule 2: Digit-sum tests — Add all the digits. If the sum is in the table of 3, the number is divisible by 3; in the table of 9, by 9.

  • Easy : 123: 1 + 2 + 3 = 6, in the table of 3, so divisible by 3.
  • Medium : 729: 7 + 2 + 9 = 18, in the table of 9, so divisible by 9 (and by 3).
  • Exam-type : 546: 5 + 4 + 6 = 15 is in the table of 3 but not of 9, so 546 is divisible by 3 but not by 9.
Remember

Divisible by both 2 and 5 means divisible by 10. Divisible by both 2 and 3 means divisible by 6.

Common mistake

Using the last digit to test 3. For 3 and 9 add all the digits.

Shortcut

Memory clue: 2-5-10 look at the tail; 3-9 add the whole body.

Spot the difference

DivisorTestExample
2last digit 0, 2, 4, 6, 8748
5last digit 0 or 51,235
3digit sum in the table of 3123
9digit sum in the table of 9729

Practice questions

Easy
  1. 1Is 748 divisible by 2?
  2. 2Is 1,235 divisible by 5?
  3. 3Is 420 divisible by 10?
Medium
  1. 4Is 123 divisible by 3?
  2. 5Is 729 divisible by 9?
  3. 6Test 546 for divisibility by 2, 3, 5 and 9.
Challenging
  1. 7Which of 4,320 and 4,321 is divisible by 2, 3, 5 and 9?
  2. 8A cook has 126 laddus. Can he share them equally among 3 children? Among 9 children?
Show answers (try them yourself first)
  1. 1Yes, it ends in 8.
  2. 2Yes, it ends in 5.
  3. 3Yes, it ends in 0.
  4. 4Yes; 1 + 2 + 3 = 6.
  5. 5Yes; 7 + 2 + 9 = 18.
  6. 6By 2: yes; by 3: yes (15); by 5: no; by 9: no.
  7. 74,320: it ends in 0 and 4 + 3 + 2 + 0 = 9.
  8. 81 + 2 + 6 = 9, so yes for both 3 and 9. 126 ÷ 3 = 42, 126 ÷ 9 = 14.

Prime and composite numbers

From daily life

A farmer wants to plant 7 saplings in equal rows. He can only do it as 1 row of 7 or 7 rows of 1. A number like this, with only two factors, is a prime number.

Simple definition

A prime number (मूळ संख्या) has exactly two factors, 1 and itself. A composite number (संयुक्त संख्या) has more than two factors. 1 is neither prime nor composite. 2 is the only even prime.

Numbers 1 to 20 with the primes 2, 3, 5, 7, 11, 13, 17 and 19 highlighted.1234567891011121314151617181920primecomposite1: neitherOnly prime even number: 2

In the figure : The primes up to 20 are shaded; 1 is neither prime nor composite.

Rule 1: Spot primes — Check whether any number other than 1 and itself divides it.

  • Easy : Primes up to 20: 2, 3, 5, 7, 11, 13, 17, 19.
  • Medium : 21 is composite because 21 = 3 × 7 has factors 1, 3, 7, 21.
  • Exam-type : Primes between 20 and 30: 23 and 29. 25, 27, 21 and 24 are composite.
Live figure

Prime and composite numbers

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Rule 2: Prime factorisation — Every composite number can be written as a product of primes.

  • Easy : 12 = 2 × 2 × 3.
  • Medium : 30 = 2 × 3 × 5.
  • Exam-type : 36 = 2 × 2 × 3 × 3; 100 = 2 × 2 × 5 × 5.
Remember

1 is not prime. 2 is the only even prime; all other even numbers are composite.

Common mistake

Calling 1 a prime number, or 9 a prime. 9 = 3 × 3, so 9 is composite.

Shortcut

Memory clue: Prime = only 1 and me.

Spot the difference

TypeNumber of factorsExample
Primeexactly 27
Compositemore than 221
Neither11

Practice questions

Easy
  1. 1Is 7 prime?
  2. 2Is 21 prime or composite?
  3. 3Write all primes up to 20.
Medium
  1. 4Which is the only even prime number?
  2. 5Is 1 prime or composite?
  3. 6Write 30 as a product of primes.
Challenging
  1. 7Write all primes between 20 and 30.
  2. 8Meena says 9 is prime because it is odd. Is she right?
Show answers (try them yourself first)
  1. 1Yes; its factors are 1 and 7 only.
  2. 2Composite, 21 = 3 × 7.
  3. 32, 3, 5, 7, 11, 13, 17, 19
  4. 42
  5. 5Neither.
  6. 62 × 3 × 5
  7. 723 and 29
  8. 8No. 9 = 3 × 3 has the factors 1, 3, 9, so it is composite.

Common multiples and common factors

From daily life

Two bells ring: one every 4 minutes, the other every 10 minutes. When will they ring together? That needs a common multiple. And if you cut two ropes into the longest equal pieces, you need a common factor.

Simple definition

A common factor divides both numbers; the biggest is the HCF (मसावि). A common multiple is a multiple of both; the smallest is the LCM (लसावि).

Two overlapping circles show the factors of 12 and 18; the overlap holds the common factors 1, 2, 3, 6, and the biggest is the HCF 6.12184129181236common: 1, 2, 3, 6 → HCF = 6

In the figure : The overlap of the factors of 12 and 18 holds the common factors 1, 2, 3, 6; the biggest is the HCF 6.

Rule 1: Common factors and HCF — List the factors of both numbers and pick the biggest one that is in both lists.

  • Easy : 12 and 18: factors of 12 are 1, 2, 3, 4, 6, 12; of 18 are 1, 2, 3, 6, 9, 18; HCF = 6.
  • Medium : 24 and 36: the common factors are 1, 2, 3, 4, 6, 12, so the HCF is 12.
  • Exam-type : Two ropes, 16 m and 40 m, are cut into the longest equal pieces: HCF of 16 and 40 is 8 m.
Live figure

Common multiples and common factors

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Rule 2: Common multiples and LCM — List the multiples of both and pick the smallest one that is in both lists.

  • Easy : 4 and 6: multiples of 4 are 4, 8, 12, 16; of 6 are 6, 12, 18; LCM = 12.
  • Medium : 6 and 8: LCM = 24.
  • Exam-type : Two bells ring every 4 and 10 minutes. LCM of 4 and 10 is 20, so they ring together every 20 minutes.
Remember

LCM answers "when together again?"; HCF answers "biggest equal groups or pieces?".

Common mistake

Mixing the two: using the HCF to find when the bells ring together. The answer must be a multiple, so use the LCM.

Shortcut

Memory clue: HCF = Highest Cut size; LCM = Least Common Meeting time.

Spot the difference

QuestionUse
when will they happen together againLCM
biggest equal groups or piecesHCF

Practice questions

Easy
  1. 1Find the HCF of 12 and 18.
  2. 2Find the LCM of 4 and 6.
  3. 3Find the HCF of 24 and 36.
Medium
  1. 4Find the LCM of 6 and 8.
  2. 5Find the HCF of 18 and 30.
  3. 6Find the LCM of 12 and 15.
Challenging
  1. 7Two ropes, 16 m and 40 m, are cut into the longest equal pieces. What is the length of each piece?
  2. 8Two bells ring every 4 and 10 minutes. If they ring together at 9:00, when will they next ring together?
Show answers (try them yourself first)
  1. 16
  2. 212
  3. 312
  4. 424
  5. 56
  6. 660
  7. 7HCF of 16 and 40 is 8 m.
  8. 8LCM of 4 and 10 is 20, so at 9:20.