Master Number System from Zero

Number System is the foundation of all aptitude. Key areas for placement: 1. Factors — how many ways to divide a number 2. Remainders — what's left after division 3. Unit digits — last digit of a power 4. Magic of 9 — patterns in powers of 9 5. Magic of 11 — palindrome number patterns These appear heavily in TCS NQT cognitive section.

Key Formulas

Number of Factors

If N = a^p × b^q × c^r, then factors = (p+1)(q+1)(r+1)

Example: 72 = 2³×3². Factors = (3+1)(2+1) = 12.

Sum of Factors

Sum = (a^(p+1)-1)/(a-1) × (b^(q+1)-1)/(b-1) × ...

Example: 12=2²×3. Sum = (2³-1)/(2-1) × (3²-1)/(3-1) = 7×4 = 28.

Remainder Rule

If N is divided by d leaving remainder r, then N = dq + r

Key: if N divided by 406 leaves 115, and 29 divides 406, then N mod 29 = 115 mod 29.

Unit Digit Cycles

Digits 2,3,7,8 have cycle of 4. Digits 4,9 have cycle of 2. Digits 0,1,5,6 always same.

Find power mod 4 to determine unit digit of 2^n, 3^n, 7^n, 8^n.

⚡ Shortcut Tricks

Number of factors of N = (p+1)(q+1)(r+1) where N = a^p × b^q × c^r. Example: 72=2³×3², factors=(3+1)(2+1)=12.

Magic of 9: 9^n always ends in 9 (odd n) or 1 (even n). 99² = 9801, 999² = 998001 — pattern: (k-1) 9s, then 8, then 0s, then 1.

Magic of 11: 11²=121, 111²=12321, 1111²=1234321 — mirror palindrome pattern.

Remainder shortcut: If N/a leaves r, and b divides a, then N/b leaves (r mod b).

Unit digit cycle: 2→2,4,8,6 (repeat 4). 3→3,9,7,1. 7→7,9,3,1. 8→8,4,2,6. Find power mod 4 to get position.

Divisibility by 9: sum of digits divisible by 9. By 11: alternate sum difference divisible by 11.

Product of factors of N = N^(number_of_factors/2). For 72: product = 72^6.

✏️ Solved Examples

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