+0

tysm

0
103
2
+227

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
+3617
+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
0

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

5^30 mod 7 =1

Sep 15, 2018