Question

If x is the smallest integer factor of 3^200 + 5^200 other than 1, what is the value of x?

A. 2
B. 3
C. 5
D. 8
E. 15

can someone explain to me how to get the answer?

Answer 1

Have you tried anything yet ?

Answer 2

uhh i dont know where to start

Answer 3

may be lets try first option

Answer 4

whats the remainder when 3 is divided by 2 ?

Answer 5

1

Answer 6

Yes, 3 leaves a remainder of 1 when divided by 2
so, 3^200 leaves a remainder of 1^200 when divided by 2

Answer 7

1^200 is same as 1
so 3^200 leaves a remainder of 1 when divided by 2

Answer 8

ohh i see

Answer 9

next work out the remainder of 5^200 when divided by 2

Answer 10

and it has to work for both values so that is the lowest one

Answer 11

3

Answer 12

whats the remainder when 5 is divided by 2 ?

Answer 13

3

Answer 14

think again

Answer 15

1

Answer 16

oops

Answer 17

:) since 5 leaves a remainder of 1 when divided by 2,
5^200 leaves a remainder of 1^200 when divided by 2

Answer 18

Overall :
3^200 + 5^200 leaves a remainder of 1+1 = 2 when divided by 2

Answer 19

but 2 is same as 0 when speaking about remainders
so 2 is the smallest integer factor

Answer 20

ahh okay

Answer 21

@ganeshie8 is right! but may i also say that no matter how many times you multiply odd numbers, that stay odd. so 3^200 is odd, so is 5^200. when you add then, that's even, i.e. divisible by 2
this particular problem can be solved in this way too :)

Answer 22

thats a very good observation and looks neat !

Answer 23

ahh okay

Answer 24

thanks!

Answer 25

yw :) are you doing number theory ?