Unit value is?

Number Theory Level 1

What is the units digit of 6 100 − 5 100 6^{100} - 5^{100} ?

None of the others 2 1 3 0

Ojasee Duble by Ojasee Duble , 19, India

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2 solutions

Jesse Nieminen
Aug 16, 2017

6 2 ≡ 10 6 ∧ 5 2 ≡ 10 5 ⟹ 6 100 ≡ 10 6 ∧ 5 100 ≡ 10 5 ⟹ 6 100 − 5 100 ≡ 10 6 − 5 ≡ 10 1 \large\begin{aligned} &6^2 \stackrel{10}\equiv 6 \ \wedge \ 5^2 \stackrel{10}\equiv 5 \\ \implies &6^{100} \stackrel{10}\equiv 6 \ \wedge \ 5^{100} \stackrel{10}\equiv 5 \\ \implies &6^{100} - 5^{100} \stackrel{10}\equiv 6 - 5 \stackrel{10}\equiv \boxed{1} \end{aligned}

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Azadali Jivani
Aug 11, 2017

unit digit of 6^100 = 6 and unit digit of 5^100 = 5
so... 6 - 5 = 1

1 Helpful 0 Interesting 0 Brilliant 0 Confused

why 600 and 500 raised to 100? in the question they are 6 and 5, right?

Ojasee Duble - 3 years, 10 months ago

thanks for indicating my mistake, dear you are right

azadali jivani - 3 years, 10 months ago

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