Mathematics and Number theory Q2

#1

(32^32^32)%7 can reduced to:

How do we do questions like these?

#3

@duttrohan0302 hey bro ,use this approach:

32^32^32=(28+4)^32^32 now if we expand this, all terms but the last one will have 28 as a multiple and thus will be divisible by 7. The last term will be 4^32^32=4^(2^5)^32=4^2^160=2^2^161=4^(2^5)^32=4^2^160=2^2^161. So we should find the remainder when 2216122161 is divided by 7.

2^1 divided by 7 yields remainder of 2;
2^2 divided by 7 yields remainder of 4;
2^3 divided by 7 yields remainder of 1;

2^4 divided by 7 yields remainder of 2;
2^5 divided by 7 yields remainder of 4;
2^6 divided by 7 yields remainder of 1;
…

The remainder repeats the pattern of 3: 2-4-1.

So we should find 2^161 (the power of 2) is 1st, 2nd or 3rd number in the above pattern of 3. 21612161 is 2 in odd power, 2 in odd power gives remainder of 2 when divided by cyclicity number 3, so it’s the second number in pattern. Which means that remainder of 2^2^161 divided by 7 would be the same as 2222 divided by 7. 2222 divided by 7 yields remainder of 4.

#4

@duttrohan0302 simply 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 a number is of the format of 4^(3k+1), it will leave a remainder of 4
If a number is of the format of 4^(3k+2) , it will leave a remainder of 2
If a number is of the format of 4^(3k) , it will leave a remainder of 1

The number given to us is 4^32^32

Let us find out Rem[Power / Cyclicity] t0 find out if it 4^(3k+1) or 4^(3k+2). We can just look at it and say that it is not 4^3k

Rem [32^32/3] = Rem [(-1)^32/3] = 1
=> The number is of the format 4^(3k + 1)
=> Rem [4^32^32 /7] = 4

#5

Rem [32^32/3] = Rem [(-1)^32/3] = 1
=> The number is of the format 4^(3k + 1)
=> Rem [4^32^32 /7] = 4

Why did you divide it by 3, and how did it become -1, are you using mod?

#6

@duttrohan0302 hey basically woh hm 1,2,3 pr check krrhe hai islie 3 lia ,also woh mod hai remainder mtlb mod lia hai usme ,also see second explanation ,it will make you motre clear.

#7

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