Question

The numbers 11284 and 7655, when divided by a certain number of three digits, leave the same remainder. Find that number of three digits?

Answer 1

Hi there!

Let us consider that three digit number = x
n' the remainder be = R

mx + R = 11284. ----(i)

nx + R = 7655. ---(ii)

[ Where, m and n are whole number multiples. ]

now,
We want to combine the equations n' eliminate R, so we'll make eqn (ii) -ve throuhout.

nx - R = -7655 

Combining :-

(m - n)x = 11284 - 7655 

(m - n)x = 3629 

∵ mx + R is greater than nx + R [ m is larger than n ]
Therefore, m - n is also a whole number.

Let this be = z. 

zx = 3629  ----(iii)

Eqn. (iii) is completely divisible by 19.
That is :-

3629 = 19 × 191 = zx

where, z = 19 n' x = 191

Here 'x' is a three-digit number.

Hence, The required answer is :-
The three digit number = 191

♦ Verification :-

i. ] 11284 / 191 = 59 ( Remainder = 15. )

ii. ] 7655 / 191 = 40 ( Remainder = 15 ) 

Hope it helps! :)