1961 AHSME Problems/Problem 28

Problem 28

If $2137^{753}$ is multiplied out, the units' digit in the final product is:

$\textbf{(A)}\ 1\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 9$

Solution

$7^1$ has a unit digit of $7$ . $7^2$ has a unit digit of $9$ . $7^3$ has a unit digit of $3$ . $7^4$ has a unit digit of $1$ . $7^5$ has a unit digit of $7$ .

Notice that the unit digit eventually cycles to itself when the exponent is increased by $4$ . It also does not matter what the other digits are in the base because the units digit is found by multiplying by only the units digit. Since $753$ leaves a remainder of $1$ after being divided by $4$ , the units digit of $2137^{753}$ is $7$ , which is answer choice $\boxed{\textbf{(D)}}$ .

SOLUTION 2

$Lemma (Fermat's Theorem)$ : If $a$ is an integer and $p$ is a prime that is prime to a the we have $a^p-1$ \equiv 1 ( $mod$ $p$ ).

Let's define $U($ x $)$ as units digit funtion of $x$ . We can clearly observe that, $U($ 7^1 $)$ = $7$

$U($ 7^4) $=$ 1 $and we can see by Fermat's Theorem that this cycle repeats with the cyclicity of$ 4 $. Now$ 753 $=$ 4k + 1 $\Rightarrow$ U( $7^753$ ) $=$ 7 $.$ ~GEOMETRY-WIZARD $

1961 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 27	Followed by Problem 29
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1961 AHSME Problems/Problem 28

Contents

Problem 28

Solution

SOLUTION 2

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