Level 5-6 Problem: The Ones Digit of the Exponent

What is the digit of ones in the number ${2003}^{2003}$?

Solution

First, let make the following observation: the ones digit in the product of two numbers always ends in the ones digit of the product of the ones digits of the factors. In other words, $1234 \times 5678$ ends in $2$ because $4 \times 8 = 32$, or $1357 \times 9753$ end in $1$ because $7 \times 3 = 21$.

Now, to raise $2003$ to the power of $2003$ means

${2003}^{2003}=\underbrace{2003\times 2003\times\ldots\times 2003}_{2003}$

Based on the definition of a exponent:
${2003}^{1}=2003$, the ones digit is $3$.
${2003}^{2}=2003 \times 2003$, the ones digit is $9$ since $3 \times 3 = 9$.
${2003}^{3}={2003}^{2} \times 2003$, the ones digit is $7$ since $9 \times 3 = 27$.
${2003}^{4}={2003}^{3} \times 2003$, the ones digit is $1$ since $7 \times 3 = 21$.
${2003}^{5}={2003}^{4} \times 2003$, the ones digit is $3$ since $1 \times 3 = 3$.
From here on, every time the power increases by $4$, the patter will repeat. Because we are raising to the power of $2003$, the patter will repeat $500$ times with three more factors of $2003$ remaining.

$2003 \div 4 = 500 r3$

Therefore,

${2003}^{2001}$ ends in $3$,
${2003}^{2002}$ ends in $9$ and
${2003}^{2003}$ ends in $7$.