134 will leave the same remainder when divided by 215,167 and 135
from 135 to 215, remainders will be different
ans 134

Rajasamantray wrote:Problem - Find the greatest number, which will divide 215,167 and 135 so as to leave the same remainder in each case.

I understood that we have to take the GCD of each number. But 167 is a prime. How do we solve this?

let k be the number which will leave same remainder 'r' in each case when divided 215,167 and 135 resp.

now, 215=k*q1+r;----1)
167=k*q2+r;----------2)
135=k*q3+r;----------3)

subtracting 2 from 1 we have;
48=k*(q1-q2);-------4)
subtracting 3 from 2 we have;
32=k*(q2-q3);--------5)
subtracting 3 from 1 we have;
80=k*(q3-q1);--------6)

k will be the hcf of 4,5,6 which is 16, hence the no. is 16...!!!

cans wrote:134 will leave the same remainder when divided by 215,167 and 135
from 135 to 215, remainders will be different
ans 134

134 leaves 81, 33 and 1 as remainder. That is not the answer. The remainder has to be the same.

Answer is 16

Rule- Greatest number with which if we divide p,q,r and it leaves same remainder in each case...
Number is of form = HCF of (p-q),(p-r)

Therefore, HCF of (215-167), (213-135) = HCF of 48, 80 = 16.

215 / 16 ---> Remainder is 7
167 / 16 ---> Remainder is 7
135 / 16 ---> Remainder is 7

Nice Explanation manpsigh!!!!