Number System

Almost every number-system question in MPSC prelims is one of four types.

1. Sum of a series. 1 + 2 + … + n = n(n+1)/2. Squares: n(n+1)(2n+1)/6.
2. Is it divisible? Use the test, never long division: 3 and 9 → add the digits; 4 → last two digits; 8 → last three; 11 → alternating sum of digits.
3. What must be added or subtracted? Divide and look at the remainder r. To subtract, remove r. To add, add divisor − r.
4. What is the units digit of a^n? The units digit of powers repeats in a cycle of at most 4. Divide n by the cycle length and use the remainder.

The single most common mistake is type 3: the paper asks what must be added and the candidate hands in the remainder. Underline the word "added" or "subtracted" before you divide — the two answers always sum to the divisor.

Remainder shortcut: if a number leaves remainder r on division by d, and f is a factor of d, the remainder on division by f is simply r ÷ f's remainder.

1+2+…+n = n(n+1)/2 · Subtract = remainder, Add = divisor − remainder · Units digit repeats in a cycle of ≤ 4

What least number must be subtracted from 5678 to make it divisible by 7? → 7 × 811 = 5677, so 5678 leaves remainder 1; subtract 1. Sanity check: 5677 ÷ 7 = 811 exactly. Had the question said "added", the answer would be 7 − 1 = 6, and 1 + 6 = 7, the divisor.