Question

In each of the following numbers, replace * by a smallest possible number to make them divisible by 11:
(a) 56*3
(b) 72 * 86
(c) 39 * 71
(d) 344 * 5​

Answer 1

Answer:

Step-by-step explanation:

Answer 2

Answer:

(a) 5643

(b) 72886

(c) 39171

(d) 34485

Step-by-step explanation:

We must know about an important concept before solving this Question,

It is about The Divisibility rule of 11,

A number is divisible by 11, if its Alternate Sum equals to 0, a Positive multiple of 11, or a Negative multiple of 11.

For ex:-

1) 7799

Here, Alternate Sum means Alternating the addition and subtraction signs,

7 - 7 + 9 - 9

= 0

Hence,

7799 is divisible by 11

2) 913

Taking Alternate sum,

9 - 1 + 3

= 11

Hence,

913 is divisible by 11

3) 2728

Taking Alternate Sum,

2 - 7 + 2 - 8

= (-11)

Hence,

2728 is divisible by 11.

Now, Let's come back to our Question,

(a) 56□3

We must fill the box with the smallest possible number to make it divisible by 11.

By divisibility of 11,

5 - 6 + □ - 3 = Multiple of 11

Let it be 11x, where x may be any integer.

So,

5 - 6 + □ - 3 = 11x

□ - 4 = 11x

Now, we can solve it only through trial and error.

Let x = 0

□ - 4 = 11(0)

□ - 4 = 0

∴ □ = 4

Hence,

The smallest number that makes it divisible by 11 is 4.

Let's check,

5 - 6 + 4 - 3

= 0

Hence, 5643 is divisible by 11.

(b) 72□86

Let's follow the similar process,

7 - 2 + □ - 8 + 6 = 11x

□ + 3 = 11x

By trial and error method,

Let x = 0

□ + 3 = 11(0)

□ + 3 = 0

□ = (-3)

But we know that,

A digit can't be negative,

So, x ≠ 0

Let x = 1

Then,

□ + 3 = 11(1)

□ + 3 = 11

□ = 11 - 3

□ = 8

Hence,

The number is 72886

Let's check,

7 - 2 + 8 - 8 + 6

= 11

Hence, it is correct.

(c) 39□71

Taking Alternate Sum,

3 - 9 + □ - 7 + 1 = 11x

□ - 12 = 11x

Let x = 0

So,

□ - 12 = 11(0)

□ - 12 = 0

□ = 12

But we know that, □ ≠ 12, as □ is a single digit of a number,

So, x ≠ 0

Let x = 1

□ - 12 = 11(1)

□ - 12 = 11

□ = 12 + 11

□ = 23

We know that, □ is a single digit, we need not take x = 2, as it would only increase the value got of □.

So,

Let x = (-1)

□ - 12 = 11(-1)

□ - 12 = (-11)

□ = 12 - 11

□ = 1

Hence,

The number is 39171.

Let's check,

3 - 9 + 1 - 7 + 1

= (-11)

Hence, 39171 is divisible by 11

(d)

344□5

Taking Alternate Sum,

3 - 4 + 4 - □ + 5 = 11x

8 - □ = 11x

Let x = 0

So,

8 - □ = 11(0)

8 - □ = 0

□ = 8

Hence,

The number is 34485.

Let's check,

3 - 4 + 4 - 8 + 5

= 0

Thus, it is divisible.

Hope it helped and believing you understood it........All the best