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#1**+3 **

1 5,

2 25,

3 125, odd

4 625, even

5 3125, odd

6 15625 even

7 78125 odd

8 390625 even

ok I have a pattern

5^137 = 1000n+ 125 = 125*8*n + 17*8 + 1 = 8(1000n+17) +1 n has a specific integer value

**Yes I made a small stupid numeric error that i am quite sure hatchet288 was smart enough to correct.**

**This last line should have been**

5^137 = 1000n+ 125 = 125*8*n + 15*8 + 5 = 8(1000n+15) +5 n has a specific integer value

so

\(5^{137}\div 8 \) has a remainder of 5

**It was no big deal, any question asker should have been able to pick up that little error for themsleves.**

**BUT Good effort Tertre, it was a good catch.**

Melody Dec 20, 2018

#2**+2 **

Melody, you're on the right track, but here is my attempt:

You're right about the odd and even switching,

5^3=125

5^4=625

5^5=last three digits of 125

5^6=last three digits are 625

So, for odd powers of five, the last three digits are 125 and for even powers five, the last three digits are 625. By the divisibility rule for 8, and we have \(\frac{125}{8}\) since 137 is odd. Thus, the remainder is \(\boxed{5}.\)

.tertre Dec 20, 2018

#4**+2 **

Here's how I view this....

5^1 / 8 = (8 - 3)^1 / 8

Every term in the expansion is a multiple of 8 except the last which is

(-3)^1 / 8 = (-3)^1 mod 8 = 5

Similarly

5^2 /8 = (8 - 3)^2 / 8

Again we only need consider the last term which is

(3)^2/8 = (3)^2 mod 8 = 1

etc....

So...we only need to consider the last term in each subsequent power expansion and note that

(3)^n mod 8 = 1 and

(-3)^n mod 8 = 5

So

5^137 / 8 = (8 - 3)^137 / 8 will have (-3)^137 / 8 as a last term

And the remainder is (-3)^137 mod 8 = 5

CPhill Dec 20, 2018

#5**0 **

Hi Chris, I expect your logic is excellent but I am having trouble following it.

You have just done the same (sort of) as Tertre and me haven't you?

I mean you have developed a pattern and used the pattern (odd and even powers) to answer the question.

I am just a little confused because you have not shown the pattern but I am assuming that is what etc means.

Melody
Dec 20, 2018