When $10^{100}$ is divided by 7, the remainder is

When dealing with huge numbers like $10^{100}$, we look for a repeating pattern in the remainders.

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Let's calculate the first few remainders when powers of $10$ are divided by $7$:

Power	Division	Remainder
$10^1$	$10 \div 7$	$3$
$10^2$	$100 \div 7$	$2$
$10^3$	$1000 \div 7$	$6$
$10^4$	$10000 \div 7$	$4$
$10^5$	$100000 \div 7$	$5$
$10^6$	$1000000 \div 7$	$1$
$10^7$	$10000000 \div 7$	$3$

Notice that $10^7$ gives remainder $3$, which is the same as $10^1$. The pattern repeats every $6$ steps: $3, 2, 6, 4, 5, 1$

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Since the pattern repeats every $6$ powers, we need to find the remainder when $100$ is divided by $6$:

$100 \div 6 = 16$ remainder $4$

This means:

$100 = 6 \times 16 + 4$

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Since $100 = 6 \times 16 + 4$, we have:

$10^{100} = 10^{6 \times 16 + 4}$

This means $10^{100}$ behaves exactly like $10^4$ in our pattern.

From our table: $10^4$ gives remainder $4$.

Therefore, the remainder when $10^{100}$ is divided by $7$ is $\boxed{4}$