Question

The greatest number which when divides 258 and 323 leaving remainders 2 and 3 respectively is
A.32
B.64
C.16
D.128

Answer 1

Hint: First find the numbers which are being completely divided from that number by deducting 2 and 3 from the given numbers respectively and in order to find the greatest such number find the common divisors of both the numbers by deduction.

Complete step-by-step answer:
Given, the greatest number let us consider it to be \[x\] which divides 258 and 323 leaving remainders 2 and 3 respectively.
Now, in order to make the 258 and 323 to be completely divisible by \[x\]we need to deduct 2 and 3 respectively from 258 and 323 respectively as then only they will give remainder 0 i.e. they would be completely divisible.
Therefore, if we deduct 2 from 258, we get 256 and similarly if we deduct 3 from 23, we get 320.
Now, we need to find the greatest number \[x\] that divides both 256 and 320.
In order to do so we need to find the greatest common divisor i.e. gcd
Now, the divisors of 256=2x2x2x2x2x2x2x2
And the divisors of 320 =2x2x2x2x2x2x5
Now the common divisors of 256 and 320 is equal to 2x2x2x2x2x2=64
Which implies the greatest number \[x\]=64
Therefore, option B) 64 is correct.

Note: Whenever we need to find the greatest number which divides the given numbers then we need to find their gcd and when one needs to find the smallest number which divides the given number then we need to find their lcm.