Official Syllabus Item: 2 Arithmetic & Number Theory

Highest Common Factor (HCF) & Lowest Common Multiple (LCM)

Q: Does the formula Product = HCF × LCM apply to three or more numbers?

No. The identity holds strictly for two numbers only. For three numbers, a × b × c != HCF × LCM.

Q: What is the rule for finding HCF and LCM of fractions?

Reduce fractions to simplest form first. HCF = (HCF of numerators) / (LCM of denominators); LCM = (LCM of numerators) / (HCF of denominators).

Q: How do you find the greatest number that divides x, y, and z leaving remainders a, b, and c?

Subtract the remainders first: HCF(x - a, y - b, z - c).

Q: How do you solve problems where the same unknown remainder is left in each case?

Compute the HCF of the pairwise positive differences: HCF(|x - y|, |y - z|, |z - x|).

Q: Why do bell tolling problems require adding 1 when finding the count of simultaneous tolls?

Because the bells toll together at starting time t = 0 as well as at each LCM interval: (T / LCM) + 1.

Q: How do you find the least number which leaves remainders r1, r2, r3 when divided by d1, d2, d3 with constant difference k?

If d - r = k is constant, Required Number = LCM(d1, d2, d3) - k.

Q: What is the minimum number of square tiles needed to pave a rectangular courtyard?

Tile side = HCF(Length, Breadth). Least number of tiles = (Length × Breadth) / [HCF(Length, Breadth)]^2.

Q: What is the HCF and LCM of two co-prime numbers?

For co-primes a and b: HCF = 1, and LCM = a × b.

Q: How does this module map to the official RIMC syllabus?

It directly fulfills Serial No. 2 of Chapter 8 in the official RIMC brochure.

Complete masterclass on HCF & LCM for RIMC Stage-I descriptive exam: prime factorization, continuous division, fractions, bell tolling, and remainder word problems.

Overview & Fundamental Concept

HCF (Greatest Common Divisor) and LCM are foundational in RIMC Stage-I descriptive mathematics. Questions range from prime factorization and Euclid's continuous division ladder to high-value 10-mark multi-step word problems involving circular tracks, simultaneous bell tolls, minimum tiles for rectangular courtyards, fractional calculations, and remainder offset theorems.

Key Theoretical Principles

HCF of given numbers is never greater than the smallest number.
LCM of given numbers is never less than the greatest number.
The product rule Product = HCF × LCM applies strictly to TWO numbers only.
Fractions must be in simplest reduced form before applying fractional HCF/LCM formulas.
In bell problems, the count of simultaneous tolls includes the starting toll at t = 0 (+1).

Illustrated Conceptual Diagrams & Visual Schematics

HCF & LCM Prime Intersection Venn Diagram & Product Law

Fundamental Law: For any two numbers $A$ and $B$, $\text{HCF}(A, B) \times \text{LCM}(A, B) = A \times B$. In the Venn diagram, $\text{HCF}$ is the product of common prime factors (intersection), while $\text{LCM}$ is the product of all prime factors (union).

Topper Strategy & Descriptive Step Breakdown

Mastering RIMC Mathematics Standard

Begin every word problem by stating why HCF or LCM is applied. Convert all measurements to uniform units (e.g. all cm or all seconds) before calculating. Conclude with a complete boxed statement specifying units.

Step 1: Given Information & Variable Declaration (1-2 Marks)

Clearly write down the data provided in the question. State any assumption (e.g., 'Let the cost price be ₹ x').

Step 2: Formula & Conceptual Foundation (1-2 Marks)

State the exact mathematical theorem or formula in standard algebraic notation before entering numeric values.

Step 3: Systematic Calculation & Reduction (2-4 Marks)

Carry out each step on a fresh line. Avoid combining three operations in one step so examiners can award partial credit.

Step 4: Final Statement with Proper Units (1 Mark)

End with a concluding statement highlighting the answer in a neat box or underline with explicit units (e.g., '₹ 1,500', '45 km/h').

Key Formulas & Conversion Cheat Sheet

Product Relation (2 Numbers)

$$a \times b = \text{HCF}(a, b) \times \text{LCM}(a, b)$$

HCF of Fractions

$$\text{HCF} = \frac{\text{HCF of Numerators}}{\text{LCM of Denominators}}$$

LCM of Fractions

$$\text{LCM} = \frac{\text{LCM of Numerators}}{\text{HCF of Denominators}}$$

Floor Tiling (Least Tiles)

$$\text{Tiles} = \frac{L \times B}{[\text{HCF}(L, B)]^2}$$

Simultaneous Tolls in Time T

$$\text{Total Tolls} = \left\lfloor \frac{T}{\text{LCM}} \right\rfloor + 1$$

Solved Model Examples (Descriptive Format with Step Mark Distribution)

Solved Model 1 Model Solution

Find the HCF of 1365, 1560, and 1755 using Euclid's continuous division method.

Step-by-Step Model Solution:

Step 1: Divide 1560 by 1365: 1560 = 1365 × 1 + 195. Then 1365 = 195 × 7 + 0 => HCF(1365, 1560) = 195.\nStep 2: Test 195 with 1755: 1755 = 195 × 9 + 0 => HCF(195, 1755) = 195.\nFinal Answer: HCF(1365, 1560, 1755) = 195.

Solved Model 2 Model Solution

Find the greatest number which divides 1305, 4665, and 6905 leaving the same remainder in each case, and find the remainder.

Step-by-Step Model Solution:

Step 1: Pairwise differences: 4665 - 1305 = 3360, 6905 - 4665 = 2240, 6905 - 1305 = 5600.\nStep 2: HCF(3360, 2240) = 1120. HCF(1120, 5600) = 1120.\nStep 3: Common remainder: 1305 = 1120 × 1 + 185.\nFinal Answer: Required number is 1120 and common remainder is 185.

Solved Model 3 Model Solution

Six bells toll together at intervals of 2, 4, 6, 8, 10, 12 seconds. In 30 minutes, how many times do they toll together?

Step-by-Step Model Solution:

Step 1: LCM(2, 4, 6, 8, 10, 12) = 120 seconds = 2 minutes.\nStep 2: In 30 minutes, number of intervals = 30 / 2 = 15.\nStep 3: Including the starting toll at t = 0: 15 + 1 = 16.\nFinal Answer: They toll together 16 times.

RIMC Practice & Drill Exercises

Attempt these questions on paper following proper step-wise conventions, then click Reveal Solution to verify your answers.

Question 1 5 Marks

Two numbers are in the ratio 15 : 11. If their HCF is 13, find the two numbers and their LCM.