HCF and LCM

Why HCF and LCM matter in AFCAT

Almost every AFCAT paper carries one or two direct questions on HCF (Highest Common Factor) and LCM (Least Common Multiple). They are pure-formula questions: no lengthy logic, no guesswork. If you know the method, you can solve each in well under a minute, which is exactly the speed AFCAT rewards in a paper where every correct answer earns three marks and every wrong one costs one.

Beyond the direct questions, HCF and LCM hide inside other chapters too — simplifying fractions, comparing ratios, finding the period of repeating events, scheduling problems and even some number-series problems. So a strong grip here pays off across the whole Numerical Ability section, not just in the one or two questions that name the topic openly.

The good news for AFCAT aspirants is that this is a closed, rule-based topic. There is nothing to interpret and no tricky wording in the core sums — once you can spot which of the two tools a question wants, the arithmetic is mechanical. That makes HCF and LCM one of the safest places to bank marks under exam pressure.

What HCF and LCM actually mean

The HCF of two or more numbers is the largest number that divides each of them exactly, leaving no remainder. For example, the HCF of 12 and 18 is 6, because 6 is the biggest number that goes into both without anything left over. The factors of 12 are 1, 2, 3, 4, 6, 12 and the factors of 18 are 1, 2, 3, 6, 9, 18; the largest factor common to both lists is 6.

The LCM is the smallest number that is a multiple of each of the given numbers — the smallest number all of them divide into exactly. The multiples of 12 are 12, 24, 36, 48… and the multiples of 18 are 18, 36, 54…; the first value appearing in both lists is 36, so the LCM of 12 and 18 is 36.

Listing factors and multiples by hand works for tiny numbers but becomes slow and error-prone for exam-sized numbers. That is why the formal methods below — prime factorisation and division — are essential for AFCAT speed.

A simple memory hook: HCF goes Down (it is a divisor, so it is smaller), LCM goes Up (it is a multiple, so it is larger). Get this direction fixed in your head and you will never confuse the two.

Method 1: Prime factorisation

Break each number into its prime factors written as powers. A prime number is one divisible only by 1 and itself — 2, 3, 5, 7, 11, 13 and so on. Every whole number can be written uniquely as a product of primes, and that is the foundation of this method. Once you have the prime forms:

• HCF = product of common primes raised to the lowest power they appear with.
• LCM = product of all primes that appear, each raised to the highest power.

Take 12 and 18. Write 12 = 2² × 3 and 18 = 2 × 3². The primes involved are 2 and 3.

This method is the safest for AFCAT when the numbers are small to medium, because it makes the logic visible on paper and avoids silly slips. It also doubles as a self-check: since both answers come from the same prime list, you can quickly confirm that your HCF divides your LCM. If you are comfortable with prime factorisation here, you will also find percentage, ratio and number-system questions easier, because they lean on the same skill.

Method 2: The division (short-division) method

For the LCM of several numbers, write them in a row and keep dividing by a prime that divides at least one of them, carrying down unchanged any number that prime does not divide. Continue until the bottom row is all 1s (or all co-prime). Multiply every left-hand divisor by the final bottom row to get the LCM. This handles three, four or more numbers neatly in one tidy table.

For the HCF of two numbers, use continuous (Euclidean) division: divide the larger by the smaller, then divide the previous divisor by the remainder, and repeat the process. The last divisor that leaves remainder 0 is the HCF. The logic is that any common factor of the two numbers must also divide their remainders, so chasing the remainders down quickly corners the largest common factor.

The magic product rule (biggest shortcut)

For exactly two numbers, there is a rule that saves enormous time and is a favourite of AFCAT examiners:

Example: two numbers are 24 and 36. Their HCF is 12. Then LCM = (24 × 36) ÷ 12 = 864 ÷ 12 = 72. No factor tree needed. The same rule run backwards lets you recover a missing number: if HCF, LCM and one number are given, the other number = (HCF × LCM) ÷ known number.

This single relationship turns many word problems into one line of division, which is exactly why it is worth memorising cold. Whenever a question gives you three of the four quantities, reach for this rule first.

HCF and LCM of fractions and decimals

AFCAT loves to test the fraction version because students forget the rule. Reduce every fraction to lowest terms first, then:

Notice the cross-pattern: HCF uses HCF-on-top, LCM-below; LCM uses LCM-on-top, HCF-below.

For decimals, first make the number of decimal places equal by adding trailing zeros, drop the decimal point so you are dealing with whole numbers, find the HCF or LCM of those whole numbers, then put the decimal point back in the result with the same number of places. For instance, to find the HCF of 0.6 and 0.18, write them as 0.60 and 0.18, treat as 60 and 18, find HCF = 6, then place two decimals back to get 0.06.

Cracking AFCAT word problems

Most marks come from word problems. The trick is recognising which tool to use:

• “Largest”, “maximum”, “greatest size”, “equal groups” → use HCF. (e.g. largest tile to pave a floor, maximum students sharing items equally.)
• “Together again”, “ring simultaneously”, “minimum number”, “least length/quantity” → use LCM. (e.g. bells tolling together, smallest number divisible by all.)

A tile-fitting version uses HCF instead: “Find the largest square tile that can pave a floor 12 m by 18 m exactly.” Here you want the biggest size that fits both dimensions, so the side of the tile is the HCF of 12 and 18, which is 6 m. The number of tiles is then (12 × 18) ÷ (6 × 6) = 6 tiles. Train your eye to translate the wording into HCF or LCM before touching any numbers — that single decision is where most marks are won or lost.

Remainder-type problems

Two classic patterns appear often. Learn both formulas.

For “greatest number that divides a, b, c leaving the same remainder”, take the HCF of the differences (a−b), (b−c), (c−a).

Speed shortcuts for the exam hall

These tricks shave precious seconds:

• Co-prime numbers (HCF = 1, like 8 and 15): their LCM is simply their product.
• One number is a factor of the other (like 6 and 24): HCF = smaller number, LCM = larger number. No working needed.
• HCF of a set always divides their LCM — if your HCF does not divide your LCM, you made an error.
• When options are given, eliminate: the correct LCM must be divisible by every number in the set; the correct HCF must divide every number.

Common mistakes that cost marks

Previous-year style question

Notice how the product rule turned a tough-looking question into one line of arithmetic — that is the AFCAT advantage.

Quick revision