The digit in the units' place of $13^{2003} - 7^{291}$ is ___.

The unit digit of a power depends only on the unit digit of the base.

So $13^{2003}$ behaves like $3^{2003}$, and powers of $3$ end in the repeating cycle $3, 9, 7, 1$ (length $4$).

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$2003 = 4 \times 500 + 3$

Remainder $3$ means the 3rd digit of the cycle.

Unit digit of $13^{2003} = 7$

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Powers of $7$ end in the repeating cycle $7, 9, 3, 1$ (length $4$).

$291 = 4 \times 72 + 3$

Remainder $3$ means the 3rd digit of the cycle.

Unit digit of $7^{291} = 3$

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$13^{2003}$ is larger than $7^{291}$, and $7 > 3$, so no borrowing is needed:

Unit digit $= 7 - 3 = 4$

Therefore, the unit digit is $4$.