Number System, HCF and LCM Notes for UPSC CSAT with Solved

The number system is the base of basic numeracy in the UPSC CSAT paper. Questions on factors, multiples, HCF, LCM, divisibility and remainders are asked directly, and the same ideas help in solving percentage, ratio, time-and-work and data interpretation questions faster. CSAT needs only Class X level mathematics, so a few rules and some practice are enough to gain marks in this area. This note explains the number system, divisibility rules, HCF and LCM with simple methods, solved examples and shortcuts, written in easy language for students from any background.

Types of Numbers

Remember that 1 is neither prime nor composite, and 2 is the only even prime number. There are 25 primes below 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97. Learn this list by heart, since it helps to test the factors of a number quickly.

Divisibility Rules

Example: Is 7854 divisible by 11? The sum of digits at the odd places (7 and 5) is 12, and at the even places (8 and 4) is 12. The difference is 0, so yes. The sum of its digits is 24, which is divisible by 3. So 7854 is divisible by 33 (3 x 11).

Factors and Multiples

A factor divides a number completely, and a multiple is obtained by multiplying the number by an integer. To find the factors, first write the number as a product of primes (prime factorisation). If a number N = a^p x b^q x c^r (where a, b and c are primes), then the number of factors = (p + 1)(q + 1)(r + 1). For example, 360 = 2³ x 3² x 5. The number of factors is (3 + 1)(2 + 1)(1 + 1) = 4 x 3 x 2 = 24. A perfect square always has an odd number of factors, while all the other numbers have an even number of factors.

HCF and LCM

HCF (Highest Common Factor) or GCD is the largest number that divides two or more numbers exactly. LCM (Least Common Multiple) is the smallest number that is exactly divisible by all of them. Use HCF when you have to divide or cut into the biggest equal parts, and use LCM when events repeat together or when you need the smallest common quantity.

Methods

• Prime factorisation: For HCF, take the lowest powers of the common primes; for LCM, take the highest powers of all primes.
• Division method: Divide by common primes step by step; the product of the divisors is the HCF. For LCM, divide by any prime that divides at least one number and multiply all divisors and the remaining numbers.
• Euclid's method for HCF: Divide the larger by the smaller, then the divisor by the remainder, and continue until the remainder is zero. The last divisor is the HCF.

Important Formulas

• For two numbers, HCF x LCM = Product of the two numbers.
• HCF of fractions = HCF of numerators / LCM of denominators.
• LCM of fractions = LCM of numerators / HCF of denominators.
• The HCF always divides the LCM; the LCM is always a multiple of the HCF.

Solved Examples

1. Two numbers have HCF 12 and LCM 180. If one number is 36, find the other. Other number = (12 x 180) / 36 = 60.
2. Three bells ring at intervals of 12, 15 and 20 minutes. They ring together at 8:00 am. When will they ring together again? LCM of 12, 15 and 20 = 60 minutes, so at 9:00 am.
3. Find the greatest number that divides 245 and 1029, leaving a remainder 5 in each case. Subtract the remainder: 240 and 1024. HCF(240, 1024) = 16.
4. Find the smallest number which, when divided by 6, 9 and 15, leaves a remainder 4 in each case. LCM of 6, 9 and 15 = 90. The number is 90 + 4 = 94.
5. Find the HCF and LCM of 2/3, 4/9 and 8/27. HCF = HCF(2, 4, 8) / LCM(3, 9, 27) = 2/27. LCM = LCM(2, 4, 8) / HCF(3, 9, 27) = 8/3.
6. What is the smallest number to be added to 1000 to make it divisible by 7? 1000 divided by 7 leaves a remainder 6, so add 1.

Remainders and Unit Digits

To find the unit digit of a large power, use the pattern of cycles. The unit digits of the powers of 2 repeat every 4 steps (2, 4, 8, 6), the powers of 3 follow 3, 9, 7, 1, the powers of 7 follow 7, 9, 3, 1, and the powers of 4 and 9 repeat in 2 steps. Digits 0, 1, 5 and 6 always keep the same unit digit. Example: the unit digit of 7⁴⁵. The cycle length is 4, and 45 divided by 4 leaves a remainder 1, so the unit digit is the first in the cycle, which is 7. If the remainder is 0, the answer is the last digit in the cycle.

Useful Sum Formulas

For example, the sum of the first 20 natural numbers is 20 x 21 / 2 = 210, and the sum of the squares of the first 10 natural numbers is 10 x 11 x 21 / 6 = 385.

Practice Questions With Answers

• Q1. Find the LCM of 8, 12 and 18. Answer: 72.
• Q2. Find the HCF of 36, 84 and 120. Answer: 12.
• Q3. What is the smallest 4-digit number divisible by 12? Answer: 1008.
• Q4. How many factors does 100 have? Answer: 9 (100 = 2 squared x 5 squared, so (2 + 1)(2 + 1) = 9).
• Q5. What is the unit digit of 3 raised to the power 22? Answer: 9.
• Q6. What is the largest 3-digit number divisible by 9? Answer: 999.

Common Mistakes to Avoid

Many students mix HCF and LCM in word problems, so before solving, ask yourself whether the question wants the "greatest" or the "least" and whether it is about splitting or about repeating. Another common error is forgetting to adjust for the remainder: in the "greatest number" problems the remainders are subtracted from the numbers, and in the "least number" problems the remainder is added at the end. Some students skip the prime factorisation and try long division, which takes more time. In the CSAT, speed matters, so practise the prime factorisation of numbers up to 100 and use the options to check your answer quickly. Finally, do not forget that the CSAT is a qualifying paper, so a few correct questions from number system topics can help you to cross the 33 per cent line safely.

Tips for CSAT

• Check the options first; with divisibility rules, you can often remove wrong options without full calculation.
• For "greatest number" word problems, think HCF; for "least number" and "together again" problems, think LCM.
• When remainders are given, subtract (for HCF) or add (for LCM) carefully.
• Practise squares up to 25 and cubes up to 10 to save time.

Revise the other basic topics with our CSAT notes and the CSAT strategy guide.

Quick Revision

• HCF x LCM = product of two numbers; number of factors = (p + 1)(q + 1)(r + 1).
• Divisibility by 11: difference of alternate digit sums is 0 or a multiple of 11.
• HCF for the greatest divisor; LCM for repeating events.
• Unit digit cycles repeat in 4 steps for 2, 3, 7 and 8.

Frequently Asked Questions (FAQ)

What is the difference between HCF and LCM?

HCF is the largest number that divides the given numbers, and LCM is the smallest number divisible by all of them.

What is the relation between HCF and LCM of two numbers?

The product of HCF and LCM equals the product of the two numbers.

How do I find the number of factors of a number?

Write the number as a product of prime powers; the number of factors is the product of each power plus one.

What is the divisibility rule of 11?

A number is divisible by 11 if the difference between the sum of digits at odd places and at even places is 0 or a multiple of 11.

What is the divisibility rule of 9?

A number is divisible by 9 if the sum of its digits is divisible by 9.

Is 1 a prime number?

No. 1 is neither prime nor composite.

Which is the only even prime number?

2 is the only even prime number.

How many prime numbers are there below 100?

There are 25 prime numbers below 100.

How do I find the HCF of fractions?

HCF of fractions equals the HCF of the numerators divided by the LCM of the denominators.

When should I use LCM in word problems?

Use LCM when events repeat together, such as bells ringing, or when you need the smallest common quantity.

When should I use HCF in word problems?

Use HCF when you need the greatest equal size or number of parts, such as cutting ropes of equal length.

How can I find the unit digit of a large power?

Use the repeating cycle of the unit digit of the base, and divide the power by the cycle length to find the position.