Find the remainder when (32^32)^32 is divided by 7.

Observe the pattern

4^1 when divided by 7, leaves a remainder of 4

4^2 when divided by 7, leaves a remainder of 2

4^3 when divided by 7, leaves a remainder of 1

And then the same cycle of 4, 2, and 1 will continue.

If the given number is of form 4^(3k+1), a remainder of 4 is obtained.

If the given number is of form 4^(3k+2), a remainder of 2 is obtained.

If the given number is of form 4^(3k), it leaves a remainder of 1.

The number given is 4^32^32

Rem [32^32/3] = Rem [(-1)^32/3] = 1

=> The number is of the form 4^(3k + 1)

=> Rem [4^32^32 /7] = 4

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