+0

# base numbers

0
239
3

For how many n, 7 <= n <= 100, ,is the base-n number 235236_n a multiple of 2?

Jul 27, 2022

#1
+1128
0

This is the "closed form" of each term:
a_n = 1/2 (6 n + (-1)^(n + 1) + 13).
And the number 235236(n) is a multiple of 7 in the following bases:
n = 10, 12, 16, 18, 22, 24, 28, 30, 34, 36, 40, 42, 46, 48, 52, 54, 58, 60, 64, 66, 70, 72, 76, 78, 82, 84, 88, 90, 94, 96, 100.

guest.

Jul 27, 2022
#3
+2666
-1

That answer checks for base-7, not base 2...

BuilderBoi  Jul 27, 2022
#2
+2666
0

If the base is odd, the number will never be even, because there are 3 odd numbers. The sum of these numbers will always be odd, because$$\text{odd} \times \text{odd} = \text{odd}$$, and $$\text{odd} + \text{odd} + \text{odd} = \text{odd}$$

However, if the base is even, all the individual digits will yield an even number because $$\text{even} \times \text{odd} = \text{even}$$, and $$\text{even} + \text{even} = \text{even}$$

So, how many even digits are there between 7 and 100?

Jul 27, 2022