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Number Systems & Computation

Mathematical Abilities — Paper-I · Official 2026 Coverage

Useful for: SSC CGL

Overview

Tier-II number questions combine divisibility, remainders, powers and factorials, and reward cyclicity and modular shortcuts.

Key points

Remainder of a product = product of remainders (then reduce).
If aᵏ leaves remainder 1 when divided by n, reduce the exponent modulo k.
Trailing zeros of n! = ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + …
Sum of first n natural numbers = n(n + 1)/2; of squares = n(n + 1)(2n + 1)/6.
A number is divisible by 11 if the alternating digit-sum difference is 0 or a multiple of 11.

Explanation

Fermat’s little theorem: for prime p not dividing a, a^(p−1) leaves remainder 1 on division by p.
Negative remainders simplify work: 99 ≡ −1 (mod 100).
Highest power of a prime p in n! = Σ ⌊n/pᵏ⌋.
Unit digit and remainder questions often combine; solve each separately.

Step by step

Identify the concept and the formula that links the given values.
Solve step by step, keeping units consistent.
Check the answer against the options and by a quick estimate.

Examples

Remainder of 2¹⁰⁰ ÷ 7: 2³ = 8 leaves 1 → 2⁹⁹ leaves 1 → 2¹⁰⁰ leaves 2.
Trailing zeros of 100!: 20 + 4 = 24.
Sum of the first 50 natural numbers = 50 × 51/2 = 1,275.

Traps and advanced points

Trap: Counting only ⌊n/5⌋ for trailing zeros.
Trap: Reducing the base but not the exponent.
Trap: Forgetting to reduce the final product of remainders.

Quick revision

aᵏ ≡ 1 → reduce exponent mod k.
Zeros = Σ ⌊n/5ᵏ⌋.
Use negative remainders.

Practice MCQs

The LCM of 12 and 18 is
1. 72
2. 6
3. 36
4. 216
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Answer: (c) 36. 12 = 2²×3 and 18 = 2×3²; LCM = 2²×3².
The HCF of 24 and 36 is
1. 12
2. 6
3. 72
4. 4
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Answer: (a) 12. The greatest common factor.
How many prime numbers are less than 20?
1. 10
2. 7
3. 9
4. 8
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Answer: (d) 8. 2, 3, 5, 7, 11, 13, 17 and 19.
The unit digit of 7⁴ is
1. 3
2. 7
3. 9
4. 1
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Answer: (d) 1. 7⁴ = 2401; unit digits of powers of 7 repeat 7, 9, 3, 1.
The smallest three-digit number divisible by 7 is
1. 107
2. 100
3. 112
4. 105
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Answer: (d) 105. 100 = 7×14 + 2, so the next multiple is 105.
1001 can be written as the product of primes
1. 3 × 7 × 47
2. 7 × 143
3. 11 × 91
4. 7 × 11 × 13
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Answer: (d) 7 × 11 × 13. 143 and 91 are not prime.

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Notes and questions are written by Kundali IQ for practice. They are not official papers; always check the latest official notification for the syllabus and pattern.