What is the 3 digit number by which when we divide 32534 and 34069, we get the same remainders

Concept:

The general formula for the division operation is-

Dividend = (Divisor × Quotient) + Remainder      ----(1)

Calculation:

Let the divisor of the numbers be d and the remainder be r.

Using (1), we get-

31513 = da + r      ----(2)

34369 = db + r      ----(3)

Here, a and b are the quotients of the numbers.

On subtracting equation (2) from equation (3), we get-

⇒ 2856 = db + r - da - r

⇒ d(b - a) = 2856

Now, let's try to make all the possible 3 digit multiples for the above equation,

⇒ d(b - a) = 2856 = 119 × 24

⇒ d(b - a) = 2856 = 238 × 12

⇒ d(b - a) = 2856 = 357 × 8

⇒ d(b - a) = 2856 = 476 × 6

⇒ d(b - a) = 2856 = 714 × 4

So, possible values of d are 119, 238, 357, 476, and 714.

Now, let's try to divide 31513 and 34369 by any of these possible values of d, we get-

⇒ \(\frac{31513}{238}=132\tfrac{97}{238}\) and \(\frac{34369}{238}=144\tfrac{97}{238}\)

⇒ 97 is the remainder.

Hence, when 31513 and 34369 are divided by a certain three digit number, then the remainder is 97.