Number System Notes for SSC Exams

Number System is the foundation of SSC Quantitative Aptitude. Two to four questions come from it in every Tier 1 paper, covering divisibility, unit digits, remainders and factors. These number system notes for SSC give you the types of numbers, all the divisibility rules and the most useful shortcuts with solved examples.

Types of Numbers

• Natural numbers: 1, 2, 3, 4 ... (counting numbers).
• Whole numbers: 0, 1, 2, 3 ... (natural numbers with zero).
• Integers: ... -2, -1, 0, 1, 2 ... (positive, negative and zero).
• Prime numbers: numbers with exactly two factors, 1 and itself (2, 3, 5, 7, 11 ...). 2 is the only even prime.
• Composite numbers: numbers with more than two factors (4, 6, 8, 9 ...). 1 is neither prime nor composite.
• Co-prime numbers: two numbers whose only common factor is 1 (8 and 15).
• Rational numbers: numbers that can be written as p/q. Irrational numbers: numbers like the square root of 2 that cannot.

The prime numbers up to 50 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47. There are 15 of them.

Divisibility Rules

Unit Digit and Cyclicity

The last digit of a power repeats after a fixed cycle. To find the unit digit of a^n, take the cycle of the unit digit of a and divide n by the cycle length (or 4).

If the remainder of n divided by 4 is 0, take the last number of the cycle.

Number of Factors and Useful Formulas

• If N = a^p x b^q x c^r (a, b, c prime), the number of factors = (p + 1)(q + 1)(r + 1).
• Sum of the first n natural numbers = n (n + 1) / 2.
• Sum of the first n odd numbers = n squared; sum of the first n even numbers = n (n + 1).
• (a + b) squared = a squared + 2ab + b squared, and a squared + b squared = (a + b) squared - 2ab.
• Dividend = Divisor x Quotient + Remainder.

Solved Examples

Example 1: Is 2,346 divisible by 6?
Solution: The last digit 6 is even, so it is divisible by 2. The digit sum 2 + 3 + 4 + 6 = 15, which is divisible by 3. So the number is divisible by 6.

Example 2: Is 7,425 divisible by 11?
Solution: Odd place digits (7 and 2) sum to 9, even place digits (4 and 5) sum to 9. The difference is 0, so it is divisible by 11 (7425 = 11 x 675).

Example 3: Find the unit digit of 7 to the power 123.
Solution: The cycle of 7 is 7, 9, 3, 1. 123 divided by 4 leaves remainder 3, so the unit digit is the third value, 3.

Example 4: Find the number of factors of 72.
Solution: 72 = 2 cubed x 3 squared. Factors = (3 + 1)(2 + 1) = 12.

Example 5: What is the smallest number to be added to 1,000 so that it is divisible by 9?
Solution: Digit sum of 1000 is 1. We need the digit sum to reach 9, so add 8 (1008 is divisible by 9).

Example 6: Find the sum of the first 50 natural numbers.
Solution: Sum = 50 x 51 / 2 = 1,275.

Quick Tips to Solve Faster

• Learn the cycles of the unit digits 2, 3, 7 and 8 by heart.
• To check divisibility by 6, 12 or 15, check the co-prime factors separately (3 and 4 for 12).
• For remainders, use the digit sum for 9 and the alternate sum for 11.
• To find the number of factors, always break the number into prime factors first.
• Use the identities (a + b) squared and (a - b) squared to avoid long calculations.

Common Mistakes to Avoid

• Treating 1 as a prime number.
• Forgetting that 2 is a prime number and is even.
• Using the wrong cycle for the unit digit when the remainder is zero.
• Applying the rule for 8 with only the last two digits.
• Counting only prime factors and not adding 1 to each power.

Quick Revision Points

• 2 is the only even prime; there are 15 primes up to 50.
• Divisibility: 9 and 3 use the digit sum; 11 uses the alternate digit sum.
• Number of factors = product of (power + 1).
• Cycle of 2 is 2, 4, 8, 6 and cycle of 3 is 3, 9, 7, 1.
• Dividend = Divisor x Quotient + Remainder.

Practice Questions With Answers

1. How many prime numbers are there between 1 and 30?
   Answer: 10 (2, 3, 5, 7, 11, 13, 17, 19, 23, 29).
2. Find the unit digit of 3 to the power 45.
   Answer: 3 (Cycle 3, 9, 7, 1; 45 divided by 4 leaves 1).
3. Find the largest 3-digit number divisible by 8.
   Answer: 992 (8 x 124).
4. Find the number of factors of 100.
   Answer: 9 (2 squared x 5 squared = 3 x 3).
5. Find the sum of the first 30 odd numbers.
   Answer: 900 (30 squared).
6. What is the remainder when 1,234 is divided by 9?
   Answer: 1 (Digit sum = 10, and 10 - 9 = 1).
7. Find the smallest 4-digit number divisible by 11.
   Answer: 1,001 (11 x 91).
8. What is the smallest number to be added to 5,000 to make it divisible by 7?
   Answer: 5 (5000 = 7 x 714 + 2).
9. Find the unit digit of 2 to the power 25.
   Answer: 2 (Cycle 2, 4, 8, 6; 25 divided by 4 leaves 1).
10. If a + b = 10 and ab = 21, find a squared + b squared.
    Answer: 58 (100 - 42).

How to Practise This Chapter

1. Learn the formulas in the table until you can write them without looking.
2. Solve the solved examples again on your own, without reading the solution.
3. Attempt the practice questions above with a timer of 10 minutes.
4. Solve the matching SSC simplification practice paper to test yourself in exam style.
5. Check the previous year papers to see how SSC frames this topic.

Related Notes and Practice Papers

• HCF and LCM Notes
• Percentage Notes
• Average Notes
• Ratio and Proportion Notes
• SSC Simplification Practice Paper

Explore More SSC Subjects

SSC exams test many subjects, so connect this topic with the related subjects below:

• Reasoning Notes
• Statistics Notes
• Indian Economy Notes
• English Notes
• General Awareness Notes
• All SSC subjects and notes

Frequently Asked Questions

What is the number system in maths?

The number system is the study of different types of numbers, such as natural numbers, integers, primes and rational numbers, and their properties.

How many questions come from number system in SSC exams?

Two to four questions usually come from number system in the Tier 1 maths section.

What is the divisibility rule of 11?

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

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

Find the cycle of the base's unit digit, divide the power by the cycle length (usually 4) and pick the digit at the remainder position.

Is 1 a prime number?

No. 1 has only one factor, so it is neither prime nor composite.

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

Write the number as a product of prime powers and multiply (power + 1) for each prime.

What is the sum of the first n natural numbers?

It is n (n + 1) / 2.

What is a co-prime pair?

Two numbers whose only common factor is 1 are co-prime, for example 8 and 15.

How should I practise number system?

Memorise the divisibility rules and cycles, solve the examples here and then attempt timed sets from previous year papers.