When 10¹⁰⁰ is divided by 7, the remainder is

The correct answer is 4.

To find the remainder when 10¹⁰⁰ is divided by 7, we can use the principles of modular arithmetic. We want to find the value of:

$${10}^{100}\equiv x(mod7)$$

First, we simplify the base by dividing 10 by 7 and finding the remainder:
$$10\equiv 3(mod7)$$

This allows us to substitute 3 for 10 in our original expression:
$${10}^{100}\equiv 3^{100}(mod7)$$

Next, we can look for a pattern in the powers of 3 modulo 7:
• 3¹ ≡ 3 (mod 7)
• 3² = 9 ≡ 2 (mod 7)
• 3³ = 27 ≡ 6 (mod 7)
• 3⁴ = 81 ≡ 4 (mod 7)
• 3⁵ = 243 ≡ 5 (mod 7)
• 3⁶ = 729 ≡ 1 (mod 7)

Alternatively, since 7 is a prime number, Fermat's Little Theorem tells us that for any integer a not divisible by 7:
$$a^{7-1}=a^6\equiv 1(mod7)$$

Therefore, we have:
$$3^6\equiv 1(mod7)$$

Now, we can express the exponent 100 in terms of multiples of 6:

Using the laws of exponents, we can rewrite the expression as:
$$3^{100}={\left(3^6\right)}^{16}\times 3^4$$

Now, applying the modular equivalence:
$$3^{100}\equiv {\left(1\right)}^{16}\times 3^4(mod7)$$

This simplifies to:
$$3^{100}\equiv 3^4(mod7)$$

Since 3⁴ = 81, we divide 81 by 7 to find the remainder:

Which means: