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# How come 30^99 + 61^100 is divisible by 31

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How come 30^99 + 61^100 is divisible by 31

Guest Aug 3, 2015

### Best Answer

#5
+81154
+10

Here's another way to prove this using the Binomial Theorem

Let (30)^99  = (31 - 1)^99

Let  (61)^100  = (62 - 1)^100

So we have

[(99C0)(31)^99  -  (99C1)(31)^98 + (99C2)(31)^97 + ....+ (99C98)(31) - 1 ]

+

[(100C0)(62)^100 - (100C1)(62)^99 + (100C2)(62)^98 + ..... +(100C98)(62)^2 - (100C99)(62) + 1 ]

And adding these, the last terms cancel, and every other term in both expressions is divisible by 31.....

CPhill  Aug 3, 2015
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#1
+14536
+10

### (30^99+61^100) modulo 31 = 0

radix  Aug 3, 2015
#2
+14536
+5

radix  Aug 3, 2015
#3
+91517
+5

Good thinking Radix.

I would like to see a proof though.  :/

Melody  Aug 3, 2015
#4
+18843
+10

How come 30^99 + 61^100 is divisible by 31

$$\small{\text{30 \equiv - 1 \pmod {31}}}\\ \small{\text{ and 61 \equiv - 1 \pmod {31}}}\\\\ \small{\text{(-1)^{99} + (-1)^{100} \stackrel{?}\equiv 0 \pmod{31}}}\\\\ \small{\text{-1 + 1 \equiv 0 \pmod{31}}}\\\\$$

heureka  Aug 3, 2015
#5
+81154
+10
Best Answer

Here's another way to prove this using the Binomial Theorem

Let (30)^99  = (31 - 1)^99

Let  (61)^100  = (62 - 1)^100

So we have

[(99C0)(31)^99  -  (99C1)(31)^98 + (99C2)(31)^97 + ....+ (99C98)(31) - 1 ]

+

[(100C0)(62)^100 - (100C1)(62)^99 + (100C2)(62)^98 + ..... +(100C98)(62)^2 - (100C99)(62) + 1 ]

And adding these, the last terms cancel, and every other term in both expressions is divisible by 31.....

CPhill  Aug 3, 2015
#6
+91517
0

Great answers.  Thanks Chris and Heureke.

I especially like yours Heureka.

I also like that Latex stacked question mark.

That'll have to find its way to the latex thread :)  I have added it  now :)

Melody  Aug 3, 2015

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