ML Aggarwal: Playing with Numbers - 3

Q.1. Which of the following numbers are divisible by 5 or by 10:
(i) 87035
(ii) 75060
(iii) 9685
(iv) 10730

We know that,
A number is divisible by 5 if its unit digit is 5 or 0.
A number is divisible by 10 if its unit digit is 0.
So, 87035, 75060, 9685, 10730 are all divisible by 5.
75060 and 10730 are divisible by 10. 

Q.2. Which of the following numbers are divisible by 2, 4 or 8:
(i) 67894
(ii) 5673244
(iii) 9685048
(iv) 6533142
(v) 75379

A number is divisible by 2 if its unit digit is 2, 4, 6, 8 or 0.
A number is divisible by 4 if the number formed by the last two digits is divisible by 4.
A number is divisible by 8 if the number formed by the last three digits is divisible by 8.
So Number 67894, 5673244, 9685048, 6533142 are divisible by 2.
Numbers, 5673244, 9685048 are divisible by 4
and numbers 9685048 is divisible by 8.

Q.3. Which of the following numbers are divisible by 3 or 9:
(i) 45639
(ii) 301248
(iii) 567081
(iv) 345903
(v) 345046

A number is divisible by 3 if the sum of its digits is divisible by 3.
A number is divisible by 9 if the sum of its digits is divisible by 9.
So the numbers 45639, 301248, 567081, 345903 are divisible by 3.
And 49639, 301248, 467081 are divisible by 9.

Q.4. Which of the following numbers are divisible by 11:
(i) 10835
(ii) 380237
(iii) 504670
(iv) 28248

A number is divisible by 11 if the difference of the sum of digits at the odd places and sum of the digits at even places is zero or divisible by 11.
So the numbers 10835, 380237, 28248 are divisible by 11.

Q.5.Which of the following numbers are divisible by 6:
(i)15414
(ii) 213888
(iii) 469876

A number is divisible by 6 if it is divisible by 2 as well as by 3.
So the numbers 15414 and 213888 are divisible by 6.

Q.6. Which of the following numbers are divisible by 7:
(i) 46f8894875
(ii) 3794856
(iii) 39823

A number is divisible by 7 if the difference of the sum of digits in alternate blocks of three digits from the right to the left is divisible by 7.
So the numbers 4618894875 and 39823 are divisible by 7.

Q.7. (i) If 34x is a multiple of 3, where x is a digit, what is the value of x?
(ii) If 74 × 5284 is a multiple of 3, where x is a digit, find the value(s) of x.

(i) If 34x is a multiple of 3, where x is a digit, what is the value of x

34x is a multiple of 3
If 3 + 4 + x = 7 + x is divisible by 3
Then x + 7 = 9
x = 9 - 7
= 2
∴ x = 2, 5, 8

(ii) If 74 × 5284 is a multiple of 3, where x is a digit, find the value(s) of x

74 × 5284 is divisible by 3
7 + 4 + x + 5 + 2 + 8 + 4 is divisible by 3
Then, 30 + x is divisible by 3
∴ x = 0, 3, 6, 9

Q.8. If 42z3 is a multiple of 9, where z is a digit, what is the value of z?

42z3 is a multiple of 9
Then, 4 + 2 + z + 3 is divisible by 9
9 + z is divisible by 9
so either 9 + z = 9 or 9 + z = 0
z = 9 + 9 = 18, or z = 9 - 9 = 0
∴ z = 0, 9

Q.9. In each of the following replace × by a digit so that the number formed is divisible by 9:
(i) 49 × 2207
(ii) 5938 × 623

(i) 49 × 2207

49 × 2207 is divisible by 9
Then, 4 + 9 + x + 2 + 2 + 0 + 7 is divisible by 9
24 + x is divisible by 9
24 + x = 27 x = 27 - 24
= 3, which is divisible by 9
∴ x = 3

(ii) 5938 × 623

5938 × 623 is divisible by 9
Then, 5 + 9 + 3 + 8 + x + 6 + 2 + 3 is divisible by 9
36 + x is divisible by 9
So, 36 + x = 36 or 45
x = 36 - 36 = 0 or x = 45 - 36 = 9
∴ x = 0, 9

Q.10. In each of the following replace * by a digit so that the number formed is divisible by 6:
(i) 97 × 542
(ii) 709 × 94

(i) 97 × 542

97 × 542
It is divisible by 6
 is divisible by 2 and 3
Since its unit digit is 2
∴ It is divisible by 2.
It is divisible by 3
Since, its sum of its digits 9 + 7 + 5 + 4 + 2 = 27 [which is divisible by 3]
27 + '*' = 27, or 30, 33, 36
∴ The '*' place can be replaced by 0 or 3 or 6 or 9.

(ii) 709 × 94

709 × 94
It is divisible by 6
It is divisible by 2 and 3
We know that its unit digit is 4
∴ It is divisible by 2
It is divisible by 3
Since its sum of its digits = 7 + 0 + 9 + 9 + 4 + * = 29 + * [which is divisible by 3]
29 + * = 30, or 33, or 36
∴ The '*' place can be replaced by 1 or 4 or 7.

Q.11. In each of the following replace * by a digit so that the number formed is divisible by 11:
(i) 64 × 2456
(ii) 86 × 6194

(i) 64 × 2456

64 × 2456
It is divisible by 11
The difference between the sum of digits of odd places and sum of digits of even place is divisible by 11or it is zero.
Now, 6 + 4 + * + 6 - 5 + 2 + 4 [which is divisible by 11]
16 + * - 11 is divisible by 11
5 + x is divisible by 11
∴ * is 6.

(ii) 86 × 6194

86 × 6194
It is divisible by 11
The difference between the sum of digits of odd places and sum of digits of even places is divisible by 11 or it is zero.
Now, 4 + 1 + * + 8 = 13 + *
9 + 6 + 6 = 21
21 - (13 + *) is divisible by 11
21 - 13 - * is divisible by 11
8 - * is divisible by 11
∴ * is 8.