AnswEr:-

Two DIGITS number = 82

Let the digit at unit's place be m & digit at ten's place be n

∴ Two digits number = m + 10n

CASE 1:-

• Two digits number is divided by sum of digits, quotient is 8 & REMAINDER is 2

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)

Case 2:-

• 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

Therefore,

Two digit number = 82 .