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How many integers \(a\) from 1 to 1000 are there such that \(a^{100}-1\) is divisible by 1000?

 

Looks like something to do with orders, I tried seperating the modulos but got stuck :<

 Sep 15, 2020
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You have to write a simple computer code to check all numbers from 1 to 1000 raised to 100 - 1. It would look like this: [a^100] - 1 mod 1000 =0. Here is a simple code that does just that:

 

a=1; p=0;c=(a^100) - 1;if(c%1000==0, goto4, goto6);printa,", ",;p=p+1; a++;if(a<=1000, goto2, 0);print"Total =",p

 

OUTPUT = 400 such numbers beween 1 and 1,000. Didn't want them listed, because they take a whole page.

If you wanted them listed, just let me know.

 Sep 15, 2020

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