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(i) Given Number = 294 

Check the divisibility by 3 

Sum of digits = 2 + 9 + 4 = 15 

15 is divisible by 3 so number 294 is also divisible by 3 

Check the divisibility by 9 . 

Sum of digits = 2 + 9 + 4 = 15 

∵ Sum 15 is not divisible by 9 

∴ 294 is not divisible by 9 

Check the divisibility by 11 

4 x 1 + 9 x (- 1) + 2 x 1 = 4 – 9 + 2 = – 3 

∴ 294 is not divisible by 11 

(ii) Given Number = 4455 

Check the divisibility by 3 

Sum of digits = 4 + 4 + 5 + 5 = 18 

18 is divisible by 3 

∴4455 is divisible by 3 

Check the divisibility by 9 

Sum of digits 18 is divisible by 9 

∴ 4455 is divisible by 9 Divisibility by 11 

5 x 1 + 5 x (-1) + 4 x 1 + 4 x (-1) 5 – 5 + 4 – 4 = 0 

∴ 4455 is divisible by 11 

(iii) Given number = 1041966 

Check the divisibility by 3 

Sum of digits = 1 + 0 + 4 + 1 + 9 + 6 + 6 = 27 

27 is divisible by 3 

So,. 1041966 is divisible by 3 

Divisibility by 9 

27 is divisible by 9 

∴1041966 is divisible by 9 

Divisibility by 11 

6 x 1 + 6 x (- 1) + 9 x 1 + 1 x (- 1) + 4 x 1 + 0 x (-1) + 1 x 1 6 – 6 + 9 – 1 + 4 + 0 + 1 

= 20 – 7 = 13 

But 13 is not divisible by 11 

So, 1041966 is not divisible by 11

Sum of digits of vertical column. 

(i) 2 + 9 + 4 = 15 

(ii) 7 + 5 + 3 = 15 

(iii) 6 + 1 + 8 = 15 

Sum of digits of horizontal row 

(i) 2 + 7 + 6 = 15 

(ii) 9 + 5 + 1 = 15 

(iii) 4 + 3 + 8 = 15 

Yes, the sum is same.

Sum of the digit = 2 + 4 + x = 6 + x 

Given Number is 24 x

Sum of the digit must be divide by 9 but it is possible if 6 + x is either 9 or 18 

If 6 + x = 9 

Then x = 9 – 6 = 3 

If 6 + x = 18, then x = 18 – 6 = 12 

Here, x is a digit, so x ≠ 12

Hence, x = 3

Given number is 89y 

∴Sum of digits = 8 + 9 + y = 17 + y 

But 17 + y is divisible by 9, this is possible only number may be either 18, 27, 36…etc. But y is a digit . 

If 17 + y = 18, then y = 18 – 17 = 1 

If 17 + y = 27, then y = 27 – 17 = 10 

But y is a digit so y ≠ 10 therefore y = 1

Given Numbers is 31M5 

∴Sum of digits = 3 + 1 + M + 5 = 9 + M 

But it is given that 9 + M is a multiple of 9 which is only possible if 9 + M is either 9 or 18. 

If 9 + M = 9 

Then M = 9 – 9 = 0 

If 9 + M = 18 

Then M = 18 – 9 = 9 

Therefore, M has two values 0 and 9

Given that R = 4 

Therefore the number becomes to 3141 

Divisibility by 11 

1 x 1 + 4 x (- 1) + 1 x 1 + 3 x (- 1)

= 1 – 4 + 1 – 3

= 2 – 7 

= – 5 

∵ – 5 is not divisible by 11 

So, 3141 or 31R1 is not divisible by 11.

Given three digit number = 24y 

∴Sum of digits = 2 + 4 + y = 6 + y 

But it is given that 6 + y is a multiple of 3, the possible values of 6 + y are 6, 9, 12, 15, …. etc. 

If 6 + y = 6, then y = 6 – 6 = 0 

If 6 + y = 9, then y = 9 – 6 = 3 

If 6 + y = 12, then y – 12 – 6 = 6 

If 6 + y = 15, then y = 15 – 6 = 9 

If 6 + y = 18, then y = 18 – 6 = 12 

But y is a digit so y ≠ 12 

Therefore, the values of y are 0, 3, 6, 9

Given Number = 31P5 

Sum of digits = 3 + 1 + P + 5 = 9 + P 

But 9 + P is a multiple of 3 so it is only possible if values of 9 + P are 9, 12, 15, 18, ….etc. 

If 9 + P = 9 then P = 0 

If 9 + P = 12 then P = 3 

If 9 + P = 15 then P = 6 

If 9 + P = 18 then P = 9 

If 9 + P = 21 then P = 12 

But P is a digit so P ≠ 12 

Therefore, values of P are 0, 3, 6 and 9

According to the rule of divisibility by 11 

= 3 × 1 + 0 × (- 1) + 0 × (1) + 9 (- 1) + 2 (1) + 6 × (- 1) + 5(1) 

= 3 + 0 + 0 – 9 + 2 – 6 + 5 

= 10 – 15 

= – 5 

That is not divisible by 11 

So, the number 5629003 is not divisible by 11

(b) 21436587

(d) ab – ba = 0

The expanded form of 37 is 37 = 10 × 3 + 7.

ab + ba = 11 (a + b) is divisible by 11.

A two digit number ab can be written as 10a + b.

(i) 10 × 5 + 6 

= 5 × 10 + 6 × 1 

= 56 

(ii) 8 × 100 + 0 × 10 + 5 

= 8 × 100 + 0 × 10 + 5 × 1 

= 805 

(iii) 9 × 100 + 9 × 10 + 9 

= 9 × 100 + 9 × 10 + 9 × 1 

= 999

(i) 42 = 4 x 10 + 2

(ii) 60 = 6 x 10 + 0

(iii) 99 = 9 x 10 + 9

(iv) 718 = 7 x 100 + 1 x 10 + 8

(i) 27 

Result 1 

Thought Number = 27 

Number obtained after reversing its digits = 72

Total of both = 27 + 72 = 99 

Dividing received number by 11 = 99/11 = 9 

Result = remainder is zero (0) . 

Result 2 

Thought Number = 27 

Number obtained after reversing its digits = 72

Subtracting small number from large number we get = 72 – 27 = 45 

Dividing received number by 9 = 45/9 = 5 

Result = remainder is zero (0) 

(ii) 67 

Result 1 

Thought Number = 67 

Number obtained after reversing its digits = 76 

Total of both = 143 

Dividing received number by 11 = 143/11 = 13 

Result = remainder is zero (0) 

Result 2

Thought Number = 67 

Number obtained after reversing its digits = 76 

Subtracting small number from large number = 76 – 67 = 9 

Dividing received number by 9 = 9/9 = 1 

Result = remainder is zero (0) 

(iii) 94 

Result 1 

Thought Number = 94 

Number obtained after reversing its digits = 49 

Total of both Number = 94 + 49 = 143 

Dividing received number by 11 = 143/11 = 13 

Result = remainder is zero (0) 

Result 2 

Thought Number = 94 

Number obtained after reversing its digits = 49 

Subtracting small number from large number = 94 – 49 = 45 

Dividing received number by 9 = 45/9 = 5 

Result = remainder is zero (0)