## 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)
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.