Evaluate 23 x 54 (mod 7)

Analyze the question: We need the remainder when 23 × 54 is divided by 7. The question could be read in two ways: as 23 × 54 in the usual base ten, or as numbers in another base. The "(mod 7)" means we want a remainder after dividing by 7.

Concept: In modular arithmetic, you can reduce each number mod 7 first, then multiply: (a × b) mod n = [(a mod n) × (b mod n)] mod n

Solution:

• 23 ÷ 7 = 3 remainder 2, so 23 ≡ 2 (mod 7)
• 54 ÷ 7 = 7 remainder 5, so 54 ≡ 5 (mod 7)

Multiply the remainders: 2 × 5 = 10, and 10 ÷ 7 = 1 remainder 3

So 23 × 54 ≡ 3 (mod 7).

Check directly: 23 × 54 = 1242. Since 7 × 177 = 1239, the remainder is 1242 − 1239 = 3. ✓

Tip: In mod questions, reduce each number first so you work with small values. Then check by doing the full multiplication if time allows.