Maths – Taking fifth roots

Calculate the fifth root of the given numbers

All results are natural numbers less than 100

Read my comment to learn how to do this in your head

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Last updated: August 18, 2019

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First submitted	August 16, 2019
Times taken	27
Average score	90.0%
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x⁵	x
7,776	6
248,832	12
45,435,424	34
102,400,000	40
11,881,376	26
3,939,040,643	83
20,511,149	29
6,240,321,451	91
90,224,199	39
1,160,290,625	65

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Level 26

Aug 16, 2019

By Euler's theorem, x⁴ ≡ 1 mod 10 and therefore x⁵ ≡ x mod 10.

This means that we can deduce the one's digit of x just by looking at the one's digit of x⁵.

To get the ten's digit, we have to guess by the size of x⁵. We ignore the last 5 digits and fins the largest 5th power that is not larger than the remaining digits. This power's fifth root will be the ten's digit of x.

Here's a useful table

1⁵ = 1

2⁵ = 32

3⁵ = 243

4⁵ = 1,024

5⁵ = 3,125

6⁵ = 7,776

7⁵ = 16,807

8⁵ = 32,768

9⁵ = 59,049

10⁵ = 100,000

Level 60