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1. In base \(b\), there are exactly one hundred three-digit numbers whose digits are all distinct. (That's "one hundred" in the ordinary sense, 100_10.)
What is b? 

 

2. What is the remainder when 5^30 is divided by 7? 

 Sep 14, 2018
edited by yasbib555  Sep 15, 2018
 #1
avatar+4396 
+2

If the digits are all distinct and there are b digits for a 3 digit number we have (we don't start numbers with the digit 0)

 

(b-1)(b-1)(b-2) = 100

 

This has 1 real solution, b=6

 

For number 2 recall Fermat's Little Theorem says \(a^{p-1}\equiv 1 \pmod{p} \text{ for p prime, a a positive integer.}\)

 

so \(5^6 \equiv 1 \pmod{7}\)

 

\(5^{30}=\left(5^6\right)^5 \equiv 1^5 \pmod{7} = 1^5 = 1\)

.
 Sep 15, 2018
 #2
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0

2. What is the remainder when 5^30 is divided by 7?

 

5^30 mod 7 =1

 Sep 15, 2018

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