13. Shortcuts and Speed Tricks

Trick 1 — Digit-sum (casting out nines) for divisibility by 3 and 9: Just add all digits repeatedly until you get a single digit; compare against 3 or 9. Example: Is 4,86,972 divisible by 9? 4+8+6+9+7+2 = 36 → 3+6 = 9 → divisible by 9 (and by 3).

Trick 2 — Divisibility by 7 using the “double-subtract” osculator (−2): Faster than long division for 4–6 digit numbers. Example: Is 1,092 divisible by 7? Last digit 2, double = 4; remaining 109; 109−4=105; last digit 5, double=10; remaining 10; 10−10=0 → divisible by 7.

Trick 3 — Divisibility by 11 via alternating sums: No need to actually divide; just alternately add and subtract digits. Example: 90,728 → (8+7+9) − (2+0) = 24 − 2 = 22 → divisible by 11.

Trick 4 — Divisibility by 13 using the “quadruple-add” osculator (+4): Example: Is 2,015 divisible by 13? Last digit 5, ×4=20; remaining 201; 201+20=221; last digit1,×4=4; remaining22; 22+4=26; 26÷13=2 exact → divisible by 13.

Trick 5 — Negative remainder method for powers: Instead of computing a full cyclicity table, check if the base is “one less than” or “one more than” the divisor. Example: Remainder of 12^50 divided by 13. Since 12 ≡ −1 (mod 13), 12^50 ≡ (−1)^50 = 1. Remainder = 1. Instant.

Trick 6 — HCF by division method for big/ugly numbers: Rather than fully factorizing large numbers (which is slow and error-prone), use Euclid’s division algorithm — it always terminates in a few steps. Example: HCF of 1,typical 4-digit pairs — shown in Section 7, Example 2.

Trick 7 — LCM via the HCF relation (for two numbers only): LCM = (Product of the two numbers) ÷ HCF. Saves you from a second full factorization once HCF is known. Example: Numbers 18 and 24. HCF = 6 (quick to spot). LCM = (18×24)/6 = 432/6 = 72.

Trick 8 — Trailing zeros in n! (factorial): Count only the number of 5’s in the prime factorization of n! (since 2’s are always more abundant than 5’s). Formula: ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + … (keep dividing by increasing powers of 5 until the quotient is 0). Example: Number of zeros at the end of 100!: ⌊100/5⌋+⌊100/25⌋+⌊100/125⌋ = 20+4+0 = 24.

Trick 9 — Sum formulas for quick series/number calculations: - Sum of first n natural numbers = n(n+1)/2 - Sum of squares of first n natural numbers = n(n+1)(2n+1)/6 - Sum of cubes of first n natural numbers = [n(n+1)/2]² Example: Sum of first 15 natural numbers = 15×16/2 = 120.

Trick 10 — Instant co-primality checks: - Any two consecutive integers are always co-prime (their HCF is always 1). - Any two distinct prime numbers are always co-prime. - A prime number and any number it does not divide are automatically co-prime. Example: 100 and 101 — instantly co-prime, no calculation needed.

Trick 11 — Compound divisibility by breaking into co-prime factors: To check divisibility by 6, 12, 15, 18, 24, etc., break the divisor into co-prime factors and check each separately (never check divisibility by a composite divisor directly using a single “combined” digit test unless it’s a standard rule like 4 or 8). Example: Check 6: split into 2 and 3 (co-prime factors of 6). Check 12: split into 4 and 3 (NOT 2 and 6, since 2 and 6 are not co-prime — always split into co-prime parts). Check 15: split into 3 and 5.

Trick 12 — Same-remainder / constant-difference number problems, solved instantly: - If a number leaves the same remainder r with every divisor in a list, the smallest such number = LCM(divisors) + r. - If (divisor − remainder) is the same constant d for every divisor in a list, the smallest such number = LCM(divisors) − d. Recognising which pattern a word problem follows (by quickly computing divisor−remainder for each pair) turns what looks like a multi-step problem into a 10-second LCM calculation. See Section 8 and Section 16 (Q. B11) for both variants worked out.

Trick 13 — Approximate square root for spotting perfect squares/primality quickly: To check if a number N is a perfect square, or to test primality by trial division, you only need to test divisibility by primes up to √N — because if N had a factor larger than √N, it would necessarily pair with a factor smaller than √N. Example: To check if 191 is prime, test only primes up to √191 ≈ 13.8, i.e., 2, 3, 5, 7, 11, 13. Since 191 is not divisible by any of these, 191 is prime — no need to check anything beyond 13.