Question

The sum of the digits of a two digit number and the number formed reversing the order of the digits is 66 , if the the two digits differ by 2, find the number. How many such numbers are there?

Answer 1

Let us assume, x and y are the two digits of the two-digit number

Therefore, the two-digit number = 10x + y and reversed number = 10y + x

Given:

10x + y + 10y + x = 66
11x + 11y = 66
x + y = 6 ---------------1

also given:

x - y = 2 --------------2

Adding equation 1 and equation 2

2x = 8
x = 4

Therefore, y = x - 2 = 4 - 2 = 2

Therefore, the two-digit number = 10x + y = 10*4 + 2 = 42.

Answer: 42 is the only such two-digit number where difference between two digits is 2 and sum of 42 and its reversed number 24 is 66

Answer 2

$\huge \underline \mathbb {SOLUTION:-}$

Let’s assume the digit at unit’s place as x and ten’s place as y. Thus from the question, the number needed to be found is 10y + x.

From the question it’s told as, the two digits of the number are differing by 2. Thus, we can write

x - y = ±2………….. (i)

Now after reversing the order of the digits, the number becomes 10x + y.

Again from the question it’s given that, the sum of the numbers obtained by reversing the digits and the original number is 66. Thus, this can be written as;

(10x+ y) + (10y+x) = 66

⇒ 10x + y + 10y + x = 66

⇒ 11x +11y = 66

⇒ 11(x + y) = 66

⇒ x + y = 66/11

⇒ x + y = 6………….. (ii)

Now, we have two sets of systems of simultaneous equations

x - y = 2 and x + y = 6

x + y = -2 and x + y = 6

Let’s first solve the first set of system of equations;

x - y = 2 …………. (iii)

x + y = 6 ………….. (iv)

On adding the equations (iii) and (iv), we get;

(x - y) + (x + y) = 2+6

⇒ x - y + x + y = 8

⇒ 2x = 8

⇒ x = 8/2

⇒ x = 4

Putting the value of x in equation (iii), we get

4 - y = 2

⇒ y = 4 – 2

⇒ y = 2

Hence:

• The required number is 10 × 2 + 4 = 24

Now, let’s solve the second set of system of equations,

x - y = -2 …………. (v)

x + y = 6 ………….. (vi)

On adding the equations (v) and (vi), we get

(x - y)+(x + y ) = -2 + 6

⇒ x - y + x + y = 4

⇒ 2x = 4

⇒ x = 4/2

⇒ x = 2

Putting the value of x in equation 5, we get;

2 - y = -2

⇒ y = 2+2

⇒ y = 4

Hence:

• The required number is 10 × 4 + 2 = 42

Therefore:

• There are two such possible numbers i.e, 24 and 42.

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