2.2 LCM, HCF & Square Roots

Why This Matters

LCM and HCF underpin problems on time intervals, tile sizes, bells tolling together, and container capacity. Square roots appear in mensuration, simplification, and number-series questions. Together these three tools account for a predictable 3 to 5 marks in the RRB Group D Mathematics section.

LCM and HCF — Definitions

• LCM (Least Common Multiple): the smallest positive number that is a multiple of each of the given numbers. For example, the LCM of 4 and 6 is 12, the smallest number both divide exactly.
• HCF (Highest Common Factor), also called GCD: the largest number that divides each of the given numbers exactly. The HCF of 12 and 18 is 6.

Two Methods

Prime factorization. Break each number into primes, then:

• LCM = product of the highest power of each prime present.
• HCF = product of the lowest power of each common prime.

Example with 24 and 36: 24 = 2^3 * 3 and 36 = 2^2 * 3^2. LCM = 2^3 * 3^2 = 72. HCF = 2^2 * 3 = 12.

Division method. Divide both numbers by common primes until co-prime remainders appear; the product of divisors is the HCF. The LCM is the HCF multiplied by the leftover co-prime factors.

The Golden Identity

For any two positive integers a and b:

LCM(a, b) times HCF(a, b) = a times b

Verify on 24 and 36: 72 * 12 = 864 and 24 * 36 = 864. The identity holds.

This identity is the fastest way to find the LCM when you know the HCF, or vice versa. It holds only for two numbers; for three or more it fails, so use prime factorization instead.

Worked example. The HCF of two numbers is 11 and their LCM is 770. If one number is 55, find the other. Using the identity: other number = (11 * 770) / 55 = 8470 / 55 = 154.

Word Problem — Bells Tolling Together

Three bells toll at intervals of 9, 12, and 15 minutes. If they toll together at 8:00 AM, when will they next toll together?

This is an LCM problem. Find LCM(9, 12, 15): 9 = 3^2, 12 = 2^2 * 3, 15 = 3 * 5. LCM = 2^2 * 3^2 * 5 = 180 minutes = 3 hours. They next toll together at 11:00 AM.

Square Roots — Three Methods

1. Prime factorization: 576 = 2^6 * 3^2, so the square root of 576 = 2^3 * 3 = 24.

2. Estimation using nearest perfect squares: estimate the square root of 680. 26^2 = 676 and 27^2 = 729, so the answer is slightly above 26. Using the linear approximation: 26 + (680 - 676)/(2 * 26) = 26 + 4/52, which is about 26.077.

3. Long-division method (for non-perfect squares, accurate to any decimal place). Pair the digits of 680 as 6.80 00 00 00 starting from the decimal point. The largest digit whose square is at most 6 is 2 (remainder 2). Bring down 80 to get 280. Double the current quotient (2 becomes 4) and find a digit d such that (40 + d) * d is at most 280; d = 6 gives 46 * 6 = 276. Quotient becomes 26. Bring down the next pair 00 and continue for more decimals.

Quick Square Root Facts

Memorising the square roots of 2 through 10 speeds up approximation questions enormously.

More Worked Examples

Example, division method for HCF. Find HCF of 72 and 120 by the division (Euclidean) method. Divide 120 by 72: 120 = 72 * 1 + 48. Divide 72 by 48: 72 = 48 * 1 + 24. Divide 48 by 24: 48 = 24 * 2 + 0. The last non-zero remainder is 24, so HCF(72, 120) = 24. Cross-check with the identity: LCM = (72 * 120) / 24 = 360. The division method is faster than prime factorisation when the numbers are large and share only small common primes.

Example, three-number LCM. Find the least length of a rope that can be cut into whole pieces of exactly 6 m, 9 m, or 15 m with nothing left over. This is LCM(6, 9, 15). Prime factors: 6 = 2 * 3, 9 = 3^2, 15 = 3 * 5. LCM = 2 * 3^2 * 5 = 90 m. Note carefully that the identity LCM * HCF = product does NOT apply to three numbers; always use prime factorisation for three or more.

Example, square root by long division to three decimals. Find the square root of 2 to three decimal places. Pair digits as 2.00 00 00 00. The largest digit whose square is at most 2 is 1 (remainder 1). Bring down 00 to get 100. Double the current quotient (1 becomes 2) and find d such that (20 + d) * d is at most 100; d = 4 gives 24 * 4 = 96 (remainder 4). Quotient is 1.4. Bring down 00 to get 400. Double the quotient ignoring the decimal (14 becomes 28) and find d such that (280 + d) * d is at most 400; d = 1 gives 281 * 1 = 281 (remainder 119). Quotient is 1.41. Bring down 00 to get 11900. Double 141 to 282 and find d such that (2820 + d) * d is at most 11900; d = 4 gives 2824 * 4 = 11296 (remainder 604). Quotient is 1.414. So the square root of 2 is approximately 1.414, matching the table above.

Example, square root of a decimal. Find the square root of 0.0144. Rewrite as 144 / 10000, so the square root is 12 / 100 = 0.12. Pairing from the decimal point also works: 0.01 44 gives the quotient 0.12 directly. The trap is to write 0.012 (dropping one decimal place) by taking the square root of 144 only and forgetting to take the square root of 10000.

Example, HCF of three numbers. Find HCF of 12, 18, and 30. Prime factors: 12 = 2^2 * 3, 18 = 2 * 3^2, 30 = 2 * 3 * 5. The only primes common to all three are 2 and 3, each raised to its lowest power across the three numbers, giving HCF = 2 * 3 = 6. Verify by division: 6 divides 12, 18, and 30 exactly, and no larger number does.

Co-prime Shortcut

For two co-prime numbers (HCF = 1), the LCM equals their product. For example, LCM(8, 9) = 72 = 8 * 9 because 8 and 9 share no common factor other than 1. This is a quick check: if the HCF of two numbers is 1, skip the working and write the product as the LCM. The same idea powers the "two buses leave together" problems where one interval is, say, 8 minutes and the other 9 minutes.

Exam Scenarios and Common Traps

• Confusing LCM with HCF: "the largest number that divides" means HCF; "the smallest number divisible by" means LCM. Read the wording carefully.
• Applying LCM times HCF equals product to three numbers: the identity fails for three or more numbers. Use prime factorization or compute pairwise.
• Forgetting units: when the LCM is in minutes, convert to hours and minutes for the final answer.
• Square root of a decimal: the square root of 0.0144 equals the square root of 144/10000 = 12/100 = 0.12. Do not drop the decimal places.

A clean grasp of these methods turns "bells tolling", "least number of tiles", and "three runners meet" questions into one-line solutions.