Number System Notes for Competitive Exams

The number system is the foundation of mathematics. Questions on divisibility, factors, HCF and LCM, remainders, unit digits and series are asked in every competitive examination, including the UPSC CSAT, SSC, banking and railway exams. Many students find this chapter tricky because it has many small rules, but it becomes simple once the rules are organised and practised. This note explains the types of numbers, divisibility tests, prime numbers, factors, HCF and LCM, remainders, unit digits and important sums, with solved examples, in simple language.

Types of Numbers

Other terms: an even number is divisible by 2, and an odd number is not. A prime number has exactly two factors, 1 and itself (2, 3, 5, 7...). 1 is neither prime nor composite, and 2 is the only even prime. A composite number has more than two factors. Co-prime numbers have HCF 1, such as 8 and 15. A perfect number equals the sum of its proper factors, like 6 and 28.

Divisibility Rules

• 2: the last digit is even. 3: the sum of digits is divisible by 3. 4: the last two digits are divisible by 4.
• 5: the last digit is 0 or 5. 6: divisible by both 2 and 3. 8: the last three digits are divisible by 8.
• 9: the sum of digits is divisible by 9. 10: the last digit is 0.
• 11: the difference between the sum of digits at odd places and the sum at even places is 0 or a multiple of 11. For 2728, (2 + 2) − (7 + 8) = −11, so it is divisible by 11 (2728 = 11 × 248).
• 7: double the last digit and subtract it from the rest. For 343, 34 − 2 × 3 = 28, which is divisible by 7, so 343 is divisible by 7.
• 12: divisible by both 3 and 4.

For a composite divisor, test with co-prime factors: a number is divisible by 18 if divisible by both 2 and 9. Do not use 2 and 6 for 12 because they are not co-prime.

Prime Numbers

The primes up to 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47, which are 15 primes. There are 25 primes below 100. To test whether a number n is prime, divide it by all primes up to the square root of n. If none divides it, it is prime. For 97, the square root is a little less than 10, so test 2, 3, 5 and 7; none divides 97, so it is prime.

Factors, Number of Factors and Sum of Factors

Write the number as a product of prime powers: N = p^a × q^b × r^c. The number of factors is (a + 1)(b + 1)(c + 1). The sum of factors is the product of (p^(a+1) − 1)/(p − 1) for each prime. Example: 360 = 2³ × 3² × 5. The number of factors is 4 × 3 × 2 = 24. The sum of factors is 15 × 13 × 6 = 1,170. A number has an odd number of factors only if it is a perfect square.

HCF and LCM

The HCF (Highest Common Factor or GCD) is the largest number that divides all the given numbers. The LCM (Least Common Multiple) is the smallest number divisible by all the given numbers. Find them by prime factorisation: HCF takes the lowest powers of common primes, and LCM takes the highest powers of all primes. The important relation for two numbers is HCF × LCM = product of the numbers. For 12 and 18, the HCF is 6 and the LCM is 36, and 6 × 36 = 216 = 12 × 18. For fractions, HCF = HCF of numerators/LCM of denominators and LCM = LCM of numerators/HCF of denominators.

Applications of HCF and LCM

The least number that is exactly divisible by 12, 15 and 20 is their LCM, which is 60. The least number that leaves a remainder 3 in each case is 60 + 3 = 63. The largest number which divides 245 and 1,029 leaving remainders 5 and 4 respectively is the HCF of (245 − 5) and (1,029 − 4), that is, HCF of 240 and 1,025, which is 5. HCF is used for cutting pieces of the greatest equal length or arranging in groups; LCM is used for events that repeat together, such as bells ringing at different intervals.

Remainders

For a product, the remainder is the product of the remainders. When 17 and 23 are each divided by 5, the remainders are 2 and 3, so the remainder of 17 × 23 is 6 divided by 5, that is, 1. (Check: 17 × 23 = 391, and 391 = 5 × 78 + 1.) The same works for addition and subtraction. If the remainder is negative, add the divisor. A useful trick is to use negative remainders: 99 divided by 10 gives remainder −1, so 99 raised to any even power leaves remainder 1.

Unit Digit and Cyclicity

The last digits of powers repeat in cycles. The digits 0, 1, 5 and 6 never change. For 2, the cycle is 2, 4, 8, 6. For 3: 3, 9, 7, 1. For 7: 7, 9, 3, 1. For 8: 8, 4, 2, 6. For 4: 4, 6. For 9: 9, 1. To find the unit digit of a^n, divide n by the cycle length (4 for 2, 3, 7 and 8) and use the remainder (if it is 0, take the last number of the cycle). For 7^95: 95 divided by 4 leaves 3, so the unit digit is the third number in the cycle of 7, which is 3. For 2^2023: 2,023 divided by 4 leaves 3, so the unit digit is 8.

Important Sums and Formulas

• Sum of first n natural numbers = n(n + 1)/2; for n = 20, it is 210.
• Sum of first n odd numbers = n²; for n = 15, it is 225.
• Sum of first n even numbers = n(n + 1).
• Sum of squares of first n natural numbers = n(n + 1)(2n + 1)/6; for n = 10, it is 385.
• Sum of cubes of first n natural numbers = [n(n + 1)/2]².

Trailing Zeros and Recurring Decimals

The number of zeros at the end of n! is the number of times 5 divides it: floor(n/5) + floor(n/25) + ... For 25!, it is 5 + 1 = 6 zeros. A recurring decimal can be written as a fraction: 0.666... = 6/9 = 2/3, and 0.272727... = 27/99 = 3/11. These ideas often appear in tricky questions.

Common Mistakes

Students call 1 a prime number, use non-co-prime factors in divisibility tests, forget to subtract the remainder before finding the HCF, or apply the remainder trick incorrectly. Always check small cases to test your method, and write the prime factorisation in a neat form.

Practice Questions With Short Solutions

Try these on your own. What is the smallest number that, when divided by 8, 12 and 18, leaves no remainder? The LCM of 8, 12 and 18 is 72. What is the remainder when 2^10 is divided by 7? Since 2³ = 8 leaves remainder 1 when divided by 7, 2^9 leaves 1 and 2^10 leaves 2. How many numbers between 1 and 100 are divisible by 7? floor(100/7) = 14. Is 1,001 divisible by 7, 11 and 13? Yes, because 1,001 = 7 × 11 × 13, which is a famous fact useful for six-digit numbers formed by repeating three digits, such as 123,123, since these are divisible by 1,001. What is the last digit of 3^2024? The cycle of 3 has length 4, and 2,024 is divisible by 4, so the unit digit is the last in the cycle, which is 1. Doing such questions daily will make you quick with patterns.

Where the Number System Is Used

This chapter is the base for many others. HCF and LCM are used in time and work, in circular tracks and in problems on bells and traffic lights. Remainders and cycles are used in calendar questions and clock problems. Divisibility rules help you check answers quickly in any calculation. Prime factorisation is needed in simplification, roots and indices. So the time you spend on this chapter will pay back in many other topics, and it also trains your mind to look for patterns, which is useful in reasoning and data interpretation.

Quick Revision

• HCF × LCM = product of two numbers; number of factors = (a + 1)(b + 1)(c + 1).
• Test primes up to the square root; 2 is the only even prime.
• Unit digit cycles of 2, 3, 7, 8 have length 4.
• Trailing zeros of n! = sum of floor(n/5^k).

The number system rewards patient practice. Learn the rules, solve a few questions of each kind every day, and soon you will see numbers as friends with patterns you can use.

Related Notes to Read Next

Continue your preparation with these related notes from the same subject:

• Average, Mixture and Alligation Notes
• Mensuration Notes for Competitive Exams
• Time, Speed and Distance Notes for Competitive Exams
• All Quantitative Aptitude notes

Frequently Asked Questions (FAQ)

What are natural numbers and whole numbers?

Natural numbers are counting numbers starting from 1, and whole numbers are natural numbers together with zero.

What is a prime number?

A prime number is a number greater than 1 that has exactly two factors, 1 and the number itself. For example, 2, 3, 5 and 7.

Is 1 a prime number?

No. The number 1 is neither prime nor composite because it has only one factor.

How do I check if a number is divisible by 11?

Find the difference between the sum of digits at odd places and the sum of digits at even places. If it is 0 or a multiple of 11, the number is divisible by 11.

What is the relation between HCF and LCM?

For two numbers, the product of their HCF and LCM equals the product of the numbers.

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

Write the number as a product of prime powers, add 1 to each power and multiply the results.

What is the HCF of fractions?

The HCF of fractions is the HCF of the numerators divided by the LCM of the denominators.

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

Use the cycle of unit digits for the base number, divide the power by the cycle length, and use the remainder to pick the digit in the cycle.

How many trailing zeros does 25! have?

25! has 6 trailing zeros, calculated as floor(25/5) + floor(25/25) = 5 + 1.

What are co-prime numbers?

Co-prime numbers are two numbers whose only common factor is 1, such as 8 and 15.

What is the sum of the first n natural numbers?

The sum is n(n + 1)/2. For n = 20, the sum is 210.

How can I convert a recurring decimal into a fraction?

Write the repeating digits over as many nines as there are digits in the repeating part. For example, 0.272727... equals 27/99, which is 3/11.