gmat6nplus1 wrote:

wizardofwashington wrote:

What is the remainder when 7^74 - 5^74 is divided by 24?

A. 0

B. 1

C. 2

D. 3

E. None of these

If you test a couple of divisions a pattern emerges

7/24 reminder 7; \(7^2\)/24 reminder 1; \(7^3\)/24 reminder 7. We can conclude that when 7^even non-negative integer, reminder is going to be 1; when 7^odd positive integer, reminder is going to be 7.

Thus \(7^7^4\) will have a reminder of 1

except for 5^1/24, which yields reminder 5. 5^2/24 yields reminder 1, 5^3/24 yields reminder 5, 5^4 yields reminder 1. Thus we can assume that 5^74 will yield reminder 1.

Now R1-R1=R0=multiple of 24.

Answer A

hi gmat6nplus1,

there are various ways to do these type of questions ..

but remember, its all about time, so very important to the easiest way .

ill just tell u three ways ..

1) just explained above by me. if u know these rules, the ans will take exactly 10 seconds..

2) as you have written by finding a pattern. may be slightly time consuming.

3) remainder theorem... for example this very Q..

mod for 24or 2^3*3 here will be=2^3*3*(1/2)(2/3)=8. It means for 24, whatever it has to divide say 'a', a^8x will be divisible by 24..

now back to the Q.. 7^74 =7^(8*9+2)= 7^(8*9)+7^2=0 +49..

similarily 5^74=5^(8*9+2)= 5^(8*9)+7^2=0 +25..

combining remainder of 7^74 - 5^74 = 49-25=24, which itself is div by 24...

here this method takes a bit longer but required where the eq is not of this form..