+0  
 
0
321
6
avatar

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

Guest Aug 3, 2015

Best Answer 

 #5
avatar+78755 
+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
Sort: 

6+0 Answers

 #1
avatar+14536 
+10

(30^99+61^100)  is divisible by 31 !

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

radix  Aug 3, 2015
 #2
avatar+14536 
+5

radix  Aug 3, 2015
 #3
avatar+91051 
+5

Good thinking Radix.   

I would like to see a proof though.  :/

Melody  Aug 3, 2015
 #4
avatar+18715 
+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
avatar+78755 
+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
avatar+91051 
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

3 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details