Number System

Remainder

Medium

The remainder when $7^{103} + 7^{101}$ is divided by $9$ is ___

Entered answer:

✅ Correct Answer: 2

Pull out the common factor:

$7^{103} + 7^{101} = 7^{101}(7^2 + 1) = 7^{101} \times 50$

Look at powers of $7$ modulo $9$ :

$7^1 \equiv 7 \pmod{9}$

$7^2 \equiv 49 \equiv 4 \pmod{9}$

$7^3 \equiv 28 \equiv 1 \pmod{9}$

So the cycle length is $3$ .

Since $101 = 3 \times 33 + 2$ :

$7^{101} \equiv 7^2 \equiv 4 \pmod{9}$

Also $50 \equiv 5 \pmod{9}$ .

$7^{101} \times 50 \equiv 4 \times 5 = 20 \equiv 2 \pmod{9}$

The remainder is $2$ .

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