## ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 5 Playing with Numbers Check Your Progress

Question 1.

In a 2-digit number, sum of digits is 7. If the difference of 2 digit number and number obtained by reversing the digits is 9, then find the number.

Solution:

Sum of a two-digit number = 7

Let unit digit = x

and ten’s digit = y

Then x + y = 7 …(i)

and number will be x + 10y

By reversing the order of the digits,

Unit digit=y

and ten’s digit = x

Then number = y + 10x

∴ (r + 10y) – (y + 10x) = 9

⇒ x + 10y – y – 10x = 9

⇒ 9y – 9x = 9 ⇒ y – x = 1 …(ii)

Adding (i) and (ii),

2y = 8 ⇒ y = 4 and x = 7 – 4 = 3

∴ Number = 3 + 10 × 4 = 3 + 40 = 43

and 4 + 10 × 3 = 4 + 30 = 34

Question 2.

In a 3 digit number, the difference of hundred’s digit and unit’s digit is 5. Find the quotient when the difference of 3-digit number and number obtained by reversing the digits is divided by 9.

Solution:

In 3-digit number,

Let unit digit = x

Ten’s digit = y

and hundreds digit = z

Now, number x + 10y + 100z

and y – x = 5 …..(i)

By reversing the digits,

Unit digit = z

Tens’d digit = y

Hundred digit = x

Then number,

⇒ z + 10y + 100x

According to the condition,

Question 3.

Without actual calculation, write the quotient when sum of 3 digit numbers 567, 675 and 756 is divided by

(i) 111

(ii) 18

(iii) 37

(iv) 3

Solution:

Sum of 3-digit of 3-digit number

= x + y + z = 5 + 6 + 7 = 18

Sum of 3-digit number = 567 + 675 + 756

(i) When divided by 111, then quotient = x + y + z = 5 + 6 + 7

(ii) When divided by 18, then quotient = 111

(iii) When divided by 37, then 3 × 18 = 54

(iv) When divided by 3, then = 37 × 18 = 666

Question 4.

Find the values of the letters in each of the following and give reasons for the steps involved:

Solution:

A = 8 – 2 = 6

B = A – 2 = 6 – 2 = 4

∴ A = 6, B = 4

3 = 7 + B ⇒ B = 3 – 7 = 13 – 7 = 6

A = 2 – B = 2 – 6 = 12 – 6 = 6 – 1 = 5

∴ A = 5, B = 6

B × B = A

∴ B = 2 × 2 = 4

and 4 × 2 = 8

Hence, A = 4, B = 2

Question 5.

If 923×783 is divisible by 11, what is the value of digit x?

Solution:

923×783 is divisible by 11

3 + 7 + 3 + 9 = 22 and 8 + x + 2 = 10 + x

Then 22 – 10 – x divisible by 11

12 – x = divisible by 11 x = 1

Question 6.

Check the divisibility of following numbers by 2, 3, 9 and 11:

(i) 76543

(ii) 65432

(iii) 98765436

(iv) 234567

Solution:

2, 3, 9, 11

(i) 76543

(a) Sum of digits = 7 + 6 + 5 + 4 + 3 = 25

∵ Unit digit is 3,

∴ It is not divisible by 2

(b) ∵ Sum of digits = 25

∴ It is not divisible by 3 as well by 9

(c) Sum of digits on odd places = 3 + 5 + 7 = 15

and on even places = 4 + 6 = 10

and difference = 15 – 10 = 5

∴ It is not divisible by 11 also.

(ii) 65432

(a) ∵ Its unit digit is 2

∴ It is divisible by 2

(b) Sum of digits = 6 + 5 + 4 + 3 + 2 = 20

So, it is not divisible by 3 as well as by 9

(c) Sum of digits at odd places = 2 + 4 + 6 = 12

and even places = 3 + 5 = 8 Difference = 12 – 8 = 4

∴ It is also not divisible by 11.

(iii) 98765436

(a) ∵ Its unit’s digit is 6

∴ It is divisible by 2

(b) Sum of digits

=6 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 48

It is divisible by 3 but not by 9

(c) Sum of digits at odd places = 6 + 4 + 6 + 8 = 24

and at even places = 3 + 5 + 7 + 9 = 24

Difference = 24 – 24 = 0

∴ It is divisible by 11.

(iv) 234567

(a) ∵ Unit digit is 7

∴ It is not divisible by 2

(b) Sum of digits = 7 + 6 + 5 + 4 + 3 + 2 = 27

∴ It is divisible by 3 as well as by 9

(c) Sum of digits at odd places = 7 + 5 + 3 = 15

and at even places = 6 + 4 + 2 = 12

Difference = 15 – 12 = 3

∴ It is not divisible by 11.

Question 7.

Check the divisibility of the following numbers by 5 or 10:

(i) 23565

(ii) 45270

Solution:

5 or 10

(i) 23565

∵ It’s unit’s digit is 5.

∴ It is divisible by 5 not by 10

(ii) 45270

∵ It’s unit’s digit is 0

∴ It is divisible by 5 as well as by 10

Question 8.

Check the divisibility of the following numbers by 4 or 8:

(i) 47596

(ii) 593024

Solution:

4 or 8

(i) 47596

(a) ∵ The number formed by its last 2-digits = 96 which is divisible by 4

∴ It is divisible by 4

(b) The number formed by it’s last 3-digits = 596

Which is not divisible by 8

∴ It is not divisible by 8

(ii) 593024

(a) ∵ The number formed by its last 2-digit = 24

Which is divisible by 4

∴ It is divisible by 4

(b) The number formed by last 3-digit = 024

Which is divisible by 8

∴ It is divisible by 8 also.