Question

Find the remainder when 2222^4444+4444^2222 us divided by 7

Answer 1

Answer:

What is the remainder when 55552222+22225555 is divided by 7 ?

The remainders when 5555 and 2222 are divided by 7 are 4 and 3 respectively.

Hence, the problem reduces to finding the remainder when

4^2222 + 3^ 5555 is divided by 7.

Now {(4)^2222 + (3)^5555}/7

= {(4^2*1111 + 3^5*1111}/7

= {(4^2)^1111 + (3^5)^1111}/7

= {(16)^1111 + (243)^1111}/7 .

This is of the form (a^n + b^n)/(a + b)

We know (a^n + b^n) is exactly divisible by (a + b) when n is ODD

Since n = 1111 ( ODD ) in this case, the expression is divisible by 16 + 243 = 259,

And 259 is a multiple of 7. ( 37 x 7 = 259)

Hence the remainder when (5555)^2222 + (2222)^5555 is divided by 7 is zero.