Question

23) When a two digits number is divided by the sum of its digits, then the

quotient is 8 and the remainder is 2. If the digits are reversed and the
resulting number so formed is divided by the sum of the digits, the
quotient is 2 and the remainder is 8. Find the two digits number.
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Answer 1

Answer: 82

Explanation:

As we know remainder formula:-

Dividend = Quotient × Divisor + Remainder

Here, dividend = Two digits number & divisor = Sum of digits

m + 10n = 8 × (m + n) + 2

m + 10n = 8m + 8n + 2

m - 8m + 10n - 8n = 2

2n - 7m = 2

2n = 7m + 2

n = (7m + 2)/2 - - (Eq.1)

Digits interchanged, two digits number is divided by sum of digits, quotient is 2 & remainder is 8.

Now, when digits interchanged:-

Two digits number = 10m + n

Using same formula we get:-

10m + n = 2 × (m + n) + 8

10m + n = 2m + 2n + 8

10m - 2m + n - 2n = 8

8m - n = 8

8m = n + 8

n = 8m - 8 - - - (Eq.2)

Comparing both equations:-

(7m + 2)/2 = 8m - 8

7m + 2 = 2(8m - 8)

7m + 2 = 16m - 16

7m - 16m = -16 - 2

-9m = -18

m = -18/-9

m = 2

So, digit at unit's place = 2

Put this value in (Eq.2)

n = 8(2) - 8

n = 16 - 8

n = 8

So, digit at ten's place = 8

Now two digit number:-

Two digit number = 2 + 10(8)

Two digit number = 2 + 80

Two digit number = 82