Playing With Numbers
Complete, step-by-step solutions for Exercises 8(A) to 8(F), divisibility tests, H.C.F., L.C.M., multiple-choice questions and reasoning questions.
Chapter Overview
This lesson organises every question into an expandable card. Open a question, study the method and compare the final answer.
Factors Made Visual
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BODMAS and Simplification
Open each question to view a clear, step-by-step solution.
Question 1Simplify: 28 - 3 × 8 ÷ 6
BODMAS order: Brackets, Of, Division, Multiplication, Addition and Subtraction.
Question 2Simplify: [(4 × 2) - (4 ÷ 2)] + 8
BODMAS order: Brackets, Of, Division, Multiplication, Addition and Subtraction.
Question 3Simplify: 15 × 12 ÷ (5 - 2)
BODMAS order: Brackets, Of, Division, Multiplication, Addition and Subtraction.
Question 4Simplify: 32 + 48 ÷ 12 - 3 × 7
BODMAS order: Brackets, Of, Division, Multiplication, Addition and Subtraction.
Question 5Simplify: 16 - 4 of 5 ÷ 10 × 2
BODMAS order: Brackets, Of, Division, Multiplication, Addition and Subtraction.
In BODMAS, of means multiplication and is evaluated before division and multiplication in this expression.
Question 6Simplify: 24 of 36 ÷ 9 - 13 × 4
BODMAS order: Brackets, Of, Division, Multiplication, Addition and Subtraction.
Question 7Simplify: 19 - [18 - {12 - (7 - 5)}]
BODMAS order: Brackets, Of, Division, Multiplication, Addition and Subtraction.
Question 8Simplify: 17 - [14 - {40 + 8 ÷ (7 - (6 - 3))}]
BODMAS order: Brackets, Of, Division, Multiplication, Addition and Subtraction.
Question 9Simplify: 25 - [12 - {5 + 18 ÷ (4 - (5 - 3))}]
BODMAS order: Brackets, Of, Division, Multiplication, Addition and Subtraction.
Question 10Simplify: 15 - [16 - {12 + 21 ÷ (9 - 2)}]
BODMAS order: Brackets, Of, Division, Multiplication, Addition and Subtraction.
Factors and Prime Numbers
Review factors, prime numbers and prime factorisation.
Question 1Write all the factors of 15, 55, 48 and 36
(i) 15: 1, 3, 5, 15
(ii) 55: 1, 5, 11, 55
(iii) 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
(iv) 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Question 2Write all the prime numbers in the given ranges
A prime number is a natural number greater than 1 that has exactly two factors: 1 and itself.
(i) Less than 25: 2, 3, 5, 7, 11, 13, 17, 19, 23
(ii) Between 15 and 35: 17, 19, 23, 29, 31
(iii) Between 8 and 76: 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73
Question 3Write the prime numbers from 5 to 45, 2 to 32 and 8 to 48
(i) From 5 to 45: 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43
(ii) From 2 to 32: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31
(iii) From 8 to 48: 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Question 4Write the prime factors of 16, 35 and 49
The only prime factor of 16 is 2.
The prime factors of 35 are 5 and 7.
The only prime factor of 49 is 7.
Highest Common Factor
Compare common-factor, prime-factor and division methods.
Question 1(i)Using the common factor method, find the H.C.F. of 25 and 20
Factors of 25: 1, 5, 25
Factors of 20: 1, 2, 4, 5, 10, 20
Common factors are 1 and 5. The highest is 5.
Question 1(ii)Using the common factor method, find the H.C.F. of 8, 12 and 18
Factors of 8: 1, 2, 4, 8
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Common factors are 1 and 2. The highest is 2.
Question 1(iii)Using the common factor method, find the H.C.F. of 24, 36, 45 and 60
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 45: 1, 3, 5, 9, 15, 45
Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Common factors are 1 and 3. The highest is 3.
Question 2(i)Using prime factorisation, find the H.C.F. of 40, 60 and 80
The common prime factors with the lowest powers are 22 and 5.
Question 2(ii)Using prime factorisation, find the H.C.F. of 48, 84 and 88
The common prime factor with the lowest power is 22.
Question 2(iii)Using prime factorisation, find the H.C.F. of 12, 16 and 28
Question 3(i)Using the division method, find the H.C.F. of 16 and 24
Divide the larger number by the smaller number. Then divide the previous divisor by the remainder.
| Dividend | Divisor | Quotient | Remainder |
|---|---|---|---|
| 24 | 16 | 1 | 8 |
| 16 | 8 | 2 | 0 |
The last non-zero divisor is 8.
Question 3(ii)Using the division method, find the H.C.F. of 7, 14 and 24
First find the H.C.F. of 7 and 14.
| Dividend | Divisor | Quotient | Remainder |
|---|---|---|---|
| 14 | 7 | 2 | 0 |
Now find the H.C.F. of 7 and 24.
| Dividend | Divisor | Quotient | Remainder |
|---|---|---|---|
| 24 | 7 | 3 | 3 |
| 7 | 3 | 2 | 1 |
| 3 | 1 | 3 | 0 |
The last non-zero divisor is 1.
Question 3(iii)Using the division method, find the H.C.F. of 32, 56 and 46
First find the H.C.F. of 32 and 56.
| Dividend | Divisor | Quotient | Remainder |
|---|---|---|---|
| 56 | 32 | 1 | 24 |
| 32 | 24 | 1 | 8 |
| 24 | 8 | 3 | 0 |
Now find the H.C.F. of 8 and 46.
| Dividend | Divisor | Quotient | Remainder |
|---|---|---|---|
| 46 | 8 | 5 | 6 |
| 8 | 6 | 1 | 2 |
| 6 | 2 | 3 | 0 |
The last non-zero divisor is 2.
Question 4(i)Using a suitable method, find the H.C.F. of 45, 75 and 135
Question 4(ii)Using a suitable method, find the H.C.F. of 66, 33 and 132
Question 4(iii)Using a suitable method, find the H.C.F. of 24, 36, 60 and 132
Question 5Find the greatest number that divides 180, 225 and 315 completely
The required greatest number is the H.C.F.
Question 6Show that 45 and 56 are co-prime numbers
Two numbers are co-prime when their H.C.F. is 1.
There is no common prime factor, so their H.C.F. is 1.
Lowest Common Multiple
Learn listing, prime-factorisation and common-division methods.
Question 1(i)Using the common multiple method, find the L.C.M. of 8, 12 and 24
Multiples of 8: 8, 16, 24, 32, 40, 48, ...
Multiples of 12: 12, 24, 36, 48, ...
Multiples of 24: 24, 48, 72, ...
The smallest common multiple is 24.
Question 1(ii)Using the common multiple method, find the L.C.M. of 10, 15 and 20
Multiples of 10: 10, 20, 30, 40, 50, 60, ...
Multiples of 15: 15, 30, 45, 60, ...
Multiples of 20: 20, 40, 60, 80, ...
The smallest common multiple is 60.
Question 1(iii)Using the common multiple method, find the L.C.M. of 3, 6, 9 and 12
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, ...
Multiples of 6: 6, 12, 18, 24, 30, 36, ...
Multiples of 9: 9, 18, 27, 36, ...
Multiples of 12: 12, 24, 36, ...
The smallest common multiple is 36.
Question 2(i)Find the L.C.M. of 18, 24 and 96 by prime factorisation and common division
Prime factor method
Common division method
| Prime divisor | Numbers after division |
|---|---|
| 2 | 9, 12, 48 |
| 2 | 9, 6, 24 |
| 2 | 9, 3, 12 |
| 2 | 9, 3, 6 |
| 2 | 9, 3, 3 |
| 3 | 3, 1, 1 |
| 3 | 1, 1, 1 |
Question 2(ii)Find the L.C.M. of 14, 21 and 98 by prime factorisation and common division
Prime factor method
Common division method
| Prime divisor | Numbers after division |
|---|---|
| 2 | 7, 21, 49 |
| 3 | 7, 7, 49 |
| 7 | 1, 1, 7 |
| 7 | 1, 1, 1 |
Question 2(iii)Find the L.C.M. of 34, 85 and 51 by prime factorisation and common division
Prime factor method
Common division method
| Prime divisor | Numbers after division |
|---|---|
| 2 | 17, 85, 51 |
| 3 | 17, 85, 17 |
| 5 | 17, 17, 17 |
| 17 | 1, 1, 1 |
Question 3Find the least number divisible by 15, 25, 40 and 50
The required number is the L.C.M.
Question 4Find the smallest number which leaves remainder 7 when divided by 18, 36 and 48
First find the L.C.M. of 18, 36 and 48.
Question 5Find the smallest number which, when increased by 11, is divisible by 16, 24, 40 and 45
First find the L.C.M. of 16, 24, 40 and 45.
Question 6Find the L.C.M. and H.C.F. of 24 and 30. Is the L.C.M. divisible by the H.C.F.?
Question 7The H.C.F. and L.C.M. are 50 and 300. One number is 150. Find the other number
Use the relation: Product of two numbers = H.C.F. × L.C.M.
Question 8The product of two numbers is 432 and their L.C.M. is 72. Find their H.C.F.
Product of two numbers = H.C.F. × L.C.M.
Question 9The product of two numbers is 19,200 and their H.C.F. is 40. Find their L.C.M.
Product of two numbers = H.C.F. × L.C.M.
Question 10Find the smallest number divisible by 12, 15, 18, 24 and 36
The required number is their L.C.M.
Divisibility Tests
Use the final digit, final two or three digits, digit sums and alternating sums.
Question 1Which of 352, 523 and 496 are divisible by 2?
A number is divisible by 2 when its last digit is even.
352 ends in 2, so it is divisible by 2.
523 ends in 3, so it is not divisible by 2.
496 ends in 6, so it is divisible by 2.
Question 2Which of 222, 532 and 678 are divisible by 4?
A number is divisible by 4 when the number formed by its last two digits is divisible by 4.
22 is not divisible by 4.
32 is divisible by 4.
78 is not divisible by 4.
Question 3Which of 324, 2536 and 92760 are divisible by 8?
A number is divisible by 8 when the number formed by its last three digits is divisible by 8.
324 is not divisible by 8.
536 ÷ 8 = 67, so 2536 is divisible by 8.
760 ÷ 8 = 95, so 92760 is divisible by 8.
Question 4Which of 221, 543 and 28492 are divisible by 3?
A number is divisible by 3 when the sum of its digits is divisible by 3.
Question 5Which of 1332, 53247 and 4968 are divisible by 9?
A number is divisible by 9 when the sum of its digits is divisible by 9.
Question 6Which of 324, 2010 and 33278 are divisible by 6?
A number is divisible by 6 when it is divisible by both 2 and 3.
324: It is even, and 3 + 2 + 4 = 9. Therefore, it is divisible by 6.
2010: It is even, and 2 + 0 + 1 + 0 = 3. Therefore, it is divisible by 6.
33278: It is even, but 3 + 3 + 2 + 7 + 8 = 23, which is not divisible by 3.
Question 7Which of 5080, 66666 and 755 are divisible by 5?
A number is divisible by 5 when its last digit is 0 or 5.
5080 ends in 0.
66666 ends in 6.
755 ends in 5.
Question 8Which of 9990, 0 and 847 are divisible by 10?
A number is divisible by 10 when its last digit is 0.
9990 ends in 0.
0 is divisible by every non-zero integer, including 10.
847 ends in 7.
Question 9Which of 5918, 68717 and 3882 are divisible by 11?
For divisibility by 11, find the difference between the sums of digits in alternating places. The difference must be 0 or a multiple of 11.
Question 10Which of 960, 8295 and 10243 are divisible by 15?
A number is divisible by 15 when it is divisible by both 3 and 5.
960: Ends in 0 and its digit sum is 15. It is divisible by 15.
8295: Ends in 5 and its digit sum is 24. It is divisible by 15.
10243: Does not end in 0 or 5. It is not divisible by 15.
Application Questions
Apply H.C.F., L.C.M. and divisibility ideas to word problems.
Question 1(i)Find the smallest number completely divisible by 28 and 42
The required number is the L.C.M.
Question 1(ii)Find the largest number that divides 28 and 42 completely
The required number is the H.C.F.
Question 2Take two numbers divisible by 8. Test whether their sum and difference are also divisible by 8
Take 16 and 24.
Both the sum and the difference are divisible by 8.
Question 3What is the H.C.F. of two consecutive numbers, consecutive even numbers and consecutive odd numbers?
(i) Consecutive numbers: H.C.F. = 1. Example: H.C.F. of 7 and 8 is 1.
(ii) Consecutive even numbers: H.C.F. = 2. Example: H.C.F. of 8 and 10 is 2.
(iii) Consecutive odd numbers: H.C.F. = 1. Example: H.C.F. of 7 and 9 is 1.
Question 4Find the smallest 3-digit number exactly divisible by 6, 8 and 12
The multiples near 100 are 96 and 120. The first 3-digit multiple is 120.
Question 5Find the greatest 3-digit number exactly divisible by 6, 8 and 12
The L.C.M. of 6, 8 and 12 is 24.
Question 6Find the L.C.M. of 140 and 168, then use it to find their H.C.F.
Use: Product of two numbers = H.C.F. × L.C.M.
Question 7Find the H.C.F. of 108 and 450, then use it to find their L.C.M.
Question 8Take any two numbers and show that the L.C.M. is divisible by the H.C.F.
Take 12 and 18.
Question 9Find the L.C.M. and H.C.F. of 16, 32 and 96. Show that the L.C.M. is divisible by the H.C.F.
Objective Practice
Check each option, then open the answer to understand the reasoning.
Question 1A number is divisible by both 5 and 8. It must be divisible by:
Because 5 and 8 are co-prime, their L.C.M. is 5 × 8 = 40.
Question 2Every number other than 0 has an infinite number of:
A non-zero number has finitely many factors but infinitely many multiples.
Question 3Find the value of 18 - (15 ÷ (12 - 9))
Question 4How many common factors do 60 and 75 have?
Common factors of 60 and 75 are 1, 3, 5 and 15. There are 4 common factors.
Question 5How many natural numbers have exactly one factor?
Only the number 1 has exactly one factor, which is 1 itself.
Question 6Which operation should be performed first in 25 × 3 + (16 ÷ 4) - 10 ÷ 2 + 3?
BODMAS requires the operation inside brackets to be completed first.
Question 7The H.C.F. and L.C.M. of 12 and 17 are respectively:
12 and 17 are co-prime. Their H.C.F. is 1 and their L.C.M. is 12 × 17.
Question 8The L.C.M. of two numbers is 56 and their H.C.F. is 4. Their product is:
Product of the two numbers = H.C.F. × L.C.M. = 4 × 56.
Question 9The H.C.F. and L.C.M. of 2 and 6 are:
Question 10The H.C.F. and L.C.M. of 7 and 15 are:
7 and 15 are co-prime. H.C.F. = 1 and L.C.M. = 7 × 15 = 105.
Statement Analysis
Evaluate each statement independently before choosing the option.
Question 11Statement 1: 4048 is divisible by 11. Statement 2: (8 + 0) - (4 + 4) = 0.
Because the alternating-sum difference is 0, 4048 is divisible by 11. Both statements are true.
Question 12Statement 1: Among 6, 12 and 216, only 216 is a multiple of 8. Statement 2: Every non-zero number has finitely many multiples.
6 and 12 are not multiples of 8, but 216 is. A non-zero number has infinitely many multiples, so Statement 2 is false.
Reasoning Practice
Use definitions and divisibility rules to test both parts.
Question 13Assertion: 1848 is divisible by 22. Reason: (8 + 8) - (4 + 1) is divisible by 11.
A number divisible by 22 must be divisible by both 2 and 11.
1848 is even, so it is divisible by 2.
The alternating-sum difference is divisible by 11, so 1848 is divisible by 11 and therefore by 22.
Question 14Assertion: 10 and 11 are co-prime. Reason: Two numbers are co-prime when their greatest common divisor is 1.
They have no common factor other than 1, so their H.C.F. is 1. The assertion and reason are both true.
Rapid Revision
Frequently Asked Questions
What is the BODMAS rule?
BODMAS gives the order for simplifying an expression: Brackets, Of, Division, Multiplication, Addition and Subtraction. Operations of equal priority are completed from left to right.
What is a factor?
A factor divides a number exactly without leaving a remainder. For example, 1, 2, 3 and 6 are factors of 6.
What is a multiple?
A multiple is obtained by multiplying a number by a whole number. For example, 6, 12, 18 and 24 are multiples of 6.
What is a prime number?
A prime number is a natural number greater than 1 with exactly two factors: 1 and the number itself.
What is H.C.F.?
H.C.F. is the greatest number that divides all the given numbers exactly. It is also called the greatest common divisor.
What is L.C.M.?
L.C.M. is the smallest positive number that is exactly divisible by all the given numbers.
What is the relation between H.C.F., L.C.M. and two numbers?
For two positive integers, the product of the numbers equals their H.C.F. multiplied by their L.C.M.
What are co-prime numbers?
Two numbers are co-prime when their H.C.F. is 1. The numbers themselves do not both need to be prime.
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