Unit's Digit=?

Algebra Level 3

If $x$ is a positive integer, what is the unit digit of $(24)^{5+2x}(36)^6(17)^3=?$

4 8 2 0 6

1 solution

Marco Brezzi
Aug 19, 2017

$\begin{aligned} 24^{5+2x}\cdot 36^6\cdot 17^3&\equiv 576^x\cdot 4^5\cdot 6^6\cdot 7^3\\ &\equiv 6^x\cdot 1024\cdot 6^6\cdot 343\\ &\equiv 6\cdot 4\cdot 6 \cdot 3 \phantom{cccccccccc}\color{#3D99F6}\text{powers of 6 always and with a 6 (see Note)}\\ &\equiv 24\cdot 18\\ &\equiv 4\cdot 8\\ &\equiv 32\equiv \boxed{2}\mod 10 \end{aligned}$

Note:

Let $x$ be a positive integer

$\begin{cases} 6^x\equiv 0^x\equiv 0 \mod 2\\ 6^x\equiv 1^x\equiv 1 \mod 5 \end{cases}$

By the CRT

$6^x\equiv 6\mod 10$

Nice solution, but there is an easier way to tackle this. Let's look at $24^{5+2x}$ , we know that powers of $4$ ends with $4$ if the power is odd and $6$ if the power is even.

Since $5+2x$ is odd $\implies$ the unit digit of $24^{5+2x}$ is $4$

Powers of $6$ always ends with $6$ and finally $7^3=343\implies 4\times 6 \times 3 = 72 \implies$ the unit

digit is $2$ .

Hana Wehbi - 3 years, 9 months ago

Unit's Digit=?

1 solution

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