Remainders & Modular Reasoning • Topic 3 of 4

Remainder Patterns

Remainders combine cleanly, which lets you replace giant numbers with their small remainders before computing. The rules: the remainder of a sum is the sum of the remainders (then reduced), and the remainder of a product is the product of the remainders (then reduced). So to find the remainder of 17 × 23 on division by 5, replace 17 by 2 and 23 by 3: 2 × 3 = 6, which leaves remainder 1. For powers, first find the remainder of the base, then look for a short cycle in the powers of that remainder. A useful special case: any power of a number ending in the same digit as the divisor pattern can collapse quickly — for division by 5, only the units digit matters (units 0 or 5 → remainder 0; units 1 or 6 → remainder 1, and so on). Reducing early is the whole trick; it keeps every number small enough for mental math.

✅ Solved examples

1. What is the remainder when 17 × 23 is divided by 5?
Reduce first: 17 ≡ 2, 23 ≡ 3 (mod 5). Product of remainders 2 × 3 = 6 ≡ 1 (mod 5). Remainder is 1.
2. What is the remainder when 2⁵⁰ is divided by 3?
2 ≡ −1 (mod 3), so 2⁵⁰ ≡ (−1)⁵⁰ = 1 (mod 3). Remainder is 1. (Or note powers of 2 mod 3 cycle 2, 1: even exponent → 1.)
3. What is the remainder when 43 + 58 + 77 is divided by 6?
Remainders: 43 ≡ 1, 58 ≡ 4, 77 ≡ 5 (mod 6). Sum 1 + 4 + 5 = 10 ≡ 4 (mod 6). Remainder is 4.
4. Quantitative Comparison. Quantity A: the remainder when 7¹⁰⁰ is divided by 10. Quantity B: the remainder when 3¹⁰⁰ is divided by 10. Which is greater?
Remainder on division by 10 is just the units digit. 7¹⁰⁰: cycle 7,9,3,1, exponent 100 ≡ 0 → units 1. 3¹⁰⁰: cycle 3,9,7,1, exponent 100 ≡ 0 → units 1. Both are 1; they are equal.

✏️ Practice — try these, take hints as needed

1. Remainder when 19 × 21 is divided by 4?
19 ≡ 3, 21 ≡ 1 (mod 4).
3 × 1 = 3.
Reduce.
3
2. Remainder when 2¹⁰⁰ is divided by 3?
2 ≡ −1 (mod 3).
(−1) to even power.
Equals 1.
1
3. Remainder when 55 + 66 + 77 is divided by 7?
55 ≡ 6, 66 ≡ 3, 77 ≡ 0.
Sum 9.
9 mod 7.
2
4. Remainder when 4²⁵ is divided by 5?
Units digit of 4²⁵.
Odd exponent → 4.
Divide by 5.
4
5. Remainder when 123 × 124 is divided by 5?
123 ≡ 3, 124 ≡ 4.
3 × 4 = 12.
12 mod 5.
2

📝 Topic test — 8 questions

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