Concept 8 of 8Foundation

📌 Formula Sheet — Prime Time

Master these five formulas — they unlock most HCF/LCM olympiad problems.

HCF × LCM = a × b

Means: for any two numbersUse when: you know HCF and product (or LCM and product) — find the missing one

Why: HCF takes each shared prime at its LOWEST power; LCM takes each prime at its HIGHEST power. Together they cover each prime exactly TWICE — which equals a × b.

⚠ Trap: This works for TWO numbers only. For three, HCF × LCM ≠ product.

Number of factors of N = (a+1)(b+1)(c+1)…

Means: if N = p^a · q^b · r^c, count factors this wayUse when: asked "how many factors does N have"

Why: A factor uses each prime 0 to a times (i.e., a+1 choices). Multiply choices across all primes.

⚠ Trap: Add 1 to each exponent BEFORE multiplying. Forgetting the +1 gives the wrong answer.

HCF via prime factors: take each COMMON prime at its LOWEST power

Means: intersection of the prime factorizationsUse when: two or more numbers

Why: A common factor must fit into BOTH numbers — so you can't take more of any prime than the smaller number has.

⚠ Trap: If a prime doesn't appear in both, it's NOT in the HCF.

LCM via prime factors: take each prime at its HIGHEST power

Means: union of prime factorizationsUse when: "smallest multiple of all given numbers"

Why: An LCM must BE a multiple of each — so it needs at least the highest power of each prime that appears anywhere.

⚠ Trap: LCM ≥ largest of the numbers, never smaller.

Co-primes: HCF(a, b) = 1

Means: share no prime factorsUse when: checking independence, simplification

Why: If two numbers share no primes, their only common factor is 1 (which divides everything).

⚠ Trap: Co-prime doesn't mean "both prime". 8 and 15 are co-prime but neither is prime.

Checking if N is prime: trial-divide up to √N

Means: only test prime divisors up to √NUse when: primality of a 3-4 digit number

Why: If N had a factor > √N, it would pair with another < √N. So if no small prime divides it, none does.

⚠ Trap: Don't stop at √N rounded DOWN if √N is not a perfect integer — include the next whole number.

💡 Tip:'When do we meet again?' → LCM. 'Largest equal pieces' → HCF.

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