Question

What is the remainder if \(19^{20}\)-\(20^{19}\) is divided by 7?

• A. 5
• B. 0
• C. 3
• D. 1
• E. 6

Answer 1

The Correct Option is A

To find the remainder of \(19^{20} - 20^{19}\) when divided by 7, we will use modular arithmetic:

Step 1: Find \(19^{20} \mod 7\). We start by reducing 19 modulo 7: \(19 \equiv 5 \pmod{7}\).

Step 2: Calculate powers of 5 modulo 7:

• \(5^1 \equiv 5 \pmod{7}\)
• \(5^2 \equiv 25 \equiv 4 \pmod{7}\)
• \(5^3 \equiv 20 \equiv 6 \pmod{7}\)
• \(5^4 \equiv 30 \equiv 2 \pmod{7}\)
• \(5^5 \equiv 10 \equiv 3 \pmod{7}\)
• \(5^6 \equiv 15 \equiv 1 \pmod{7}\)

Since \(5^6 \equiv 1 \pmod{7}\), we can reduce the power 20 modulo 6: \(20 \equiv 2 \pmod{6}\). Hence, \(5^{20} \equiv 5^2 \equiv 4 \pmod{7}\).

Step 3: Find \(20^{19} \mod 7\). We reduce 20 modulo 7: \(20 \equiv 6 \pmod{7}\).

Step 4: Calculate powers of 6 modulo 7:

• \(6^1 \equiv 6 \pmod{7}\)
• \(6^2 \equiv 36 \equiv 1 \pmod{7}\)

Since \(6^2 \equiv 1 \pmod{7}\), we reduce the power 19 modulo 2: \(19 \equiv 1 \pmod{2}\). Therefore, \(6^{19} \equiv 6^1 \equiv 6 \pmod{7}\).

Step 5: Compute the remainder of \(19^{20} - 20^{19}\) modulo 7: \((5^{20} - 6^{19}) \equiv (4 - 6) \equiv -2 \equiv 5 \pmod{7}\).

Thus, the remainder is 5.