Number System Fundamentals • Topic 3 of 7

Remainders

Remainder questions reward modular thinking. If a number leaves remainder r on division by d, it has the form d × k + r. To find the remainder of a large product or power, take the remainder of each part first, then multiply or raise to the power and reduce again. When a number leaves the same remainder r for several divisors, it has the form LCM(divisors) × k + r. For powers, look for a small power that leaves remainder 1 — then the exponent can be reduced modulo that cycle length. These shortcuts answer "what is left over" problems without heavy arithmetic.

✅ Solved examples

1. A number leaves remainder 5 when divided by 7 and remainder 7 when divided by 9. Find the smallest such number.
List numbers that are 7 more than a multiple of 9: 7, 16, 25, 34, 43, 52, 61. Test each for remainder 5 on division by 7: 61 = 7 × 8 + 5. So the smallest number is 61.
2. What is the remainder when 7¹⁰⁰ is divided by 5?
7 leaves remainder 2 on division by 5, so we need 2¹⁰⁰ mod 5. Since 2⁴ = 16 leaves remainder 1, and 100 = 4 × 25, 2¹⁰⁰ leaves remainder 1.
3. Find the remainder when 17 × 23 is divided by 5.
17 leaves remainder 2 and 23 leaves remainder 3. Multiply the remainders: 2 × 3 = 6, which leaves remainder 1 on division by 5.
4. What is the greatest three-digit number that leaves remainder 1 when divided by 4, 5 and 6?
The number has the form LCM(4,5,6) × k + 1 = 60k + 1. The largest such value under 1000 is 60 × 16 + 1 = 961.

✏️ Practice — try these, take hints as needed

1. What is the remainder when 2⁵⁰ is divided by 7?
Find a small power of 2 that leaves remainder 1 mod 7.
2³ = 8 leaves remainder 1, so the cycle length is 3.
Write 50 = 3 × 16 + 2 and use 2² = 4.
4.
2. Find the least number greater than 1 that leaves remainder 1 when divided by both 6 and 8.
Same remainder means LCM(divisors) × k + 1.
LCM(6, 8) = 24.
Take k = 1.
25.
3. Find the remainder when 12 × 13 × 14 is divided by 5.
Replace each factor by its remainder mod 5.
12 → 2, 13 → 3, 14 → 4.
Multiply 2 × 3 × 4 = 24 and reduce mod 5.
4.
4. A number divided by 5 gives quotient 6 and remainder 3. What is the number?
Use number = divisor × quotient + remainder.
Substitute divisor 5, quotient 6, remainder 3.
Compute 5 × 6 + 3.
33.
5. What is the remainder when 3²⁰ is divided by 8?
Find a small power of 3 that leaves remainder 1 mod 8.
3² = 9 leaves remainder 1.
Since 20 is even, 3²⁰ = (3²)¹⁰.
1.

📝 Topic test — 8 questions

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