+0

# Another mod equation

0
139
3

Here is another modular equation. What is the remainder of: 17^1507 mod 102? Will appreciate any help. Thank you.

Dec 20, 2018

#1
+1

I will give the answer that my calculator gives and then one of you mathematicians can break it down into smaller parts, if that is possible: 17^1507 mod 102 =17.

Dec 21, 2018
#2
+100802
+2

Here is another modular equation. What is the remainder of: 17^1507 mod 102? Will appreciate any help. Thank you.

102=6*17

17^1507 = 0 (mod17)

so  17^1507 (mod 102)  will be   on of these.    0,17,34,51,68 or 85

17^1507 (mod 6) = (-1)^1507 (mod 6) = -1

So which of the above possibilities will equal -1 (or 5 which is the same) in Mod 6

The only answer that works is 17

Dec 21, 2018
#3
+5074
+1

$$\text{Using Euler's Theorem}\\ 17^{\varphi(102)}\equiv 1 \pmod{102}\\ \varphi(102) = 32\\ 1507 = 32 \cdot 47 + 3\\ 17^{1507} = 17^{32 \cdot 47 + 3} = 17^{32\cdot 47}17^3 \equiv 17^3 \pmod{102}\\ \text{this we can solve by brute force w/o too much trouble}\\ 17^3 = 4913 = 48\cdot 102 + 17 \equiv 17 \pmod{ 102}$$

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Dec 21, 2018