SSC GD Number System Notes 2026 – Complete Notes, Formulas & Solved Questions

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SSC GD Number System

Detailed Handwritten Notes • Very Simple English • 2026

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SSC GD Number System Notes – Complete Chapter

Easy idea: A number tells us how many, how much, or where something is placed. In SSC GD Maths, Number System questions mainly test simple rules about numbers. Learn the rule, understand one example, and then practise.

1. Basic Number Types

Natural Numbers

Natural numbers are counting numbers.

\[N=\{1,2,3,4,5,\ldots\}\]

Whole Numbers

Whole numbers are natural numbers together with zero.

\[W=\{0,1,2,3,4,\ldots\}\]

Integers

Integers include negative numbers, zero and positive numbers.

\[Z=\{\ldots,-3,-2,-1,0,1,2,3,\ldots\}\]

Rational Numbers

A rational number can be written in the form \(\frac pq\), where \(p\) and \(q\) are integers and \(q\ne0\). Examples: \(\frac35\), \(-\frac72\), 4 and 0.

Irrational Numbers

An irrational number cannot be written in the form \(\frac pq\). Examples: \(\sqrt2\), \(\sqrt3\), \(\pi\).

Prime Numbers

A prime number has exactly two positive factors: 1 and itself. Examples: 2, 3, 5, 7, 11.

Important: 2 is the only even prime number.

Composite Numbers

A composite number has more than two positive factors. Examples: 4, 6, 8, 9, 10.

Co-Prime Numbers

Two numbers are co-prime when their HCF is 1.

\[\operatorname{HCF}(a,b)=1\]

Even and Odd Numbers

An even number is exactly divisible by 2. An odd number is not exactly divisible by 2.

\[\text{Even}=2n\]\[\text{Odd}=2n+1\]

Successor and Predecessor

Successor means the next number. Predecessor means the previous number.

\[\text{Successor of }n=n+1\]\[\text{Predecessor of }n=n-1\]

Consecutive Numbers

Numbers that come one after another are called consecutive numbers.

\[n,\ n+1,\ n+2,\ n+3\]

2. Important Number Properties

Positive and Negative Numbers

Positive numbers are greater than zero. Negative numbers are less than zero.

Sign Rules

\[(+)\times(+)=+\]\[(-)\times(-)=+\]\[(+)\times(-)=-\]\[(-)\times(+)=-\]

Useful Identities

\[(a+b)^2=a^2+2ab+b^2\]\[(a-b)^2=a^2-2ab+b^2\]\[a^2-b^2=(a-b)(a+b)\]

Sum Formulas

\[1+2+3+\cdots+n=\frac{n(n+1)}2\]\[1+3+5+\cdots+(2n-1)=n^2\]\[2+4+6+\cdots+2n=n(n+1)\]

3. Divisibility Rules

Divisor	Easy Rule	Example
2	Last digit is 0,2,4,6,8.	248
3	Sum of digits is divisible by 3.	123 → 6
4	Last two digits are divisible by 4.	316 → 16
5	Last digit is 0 or 5.	245
6	Divisible by both 2 and 3.	258
8	Last three digits are divisible by 8.	312
9	Digit sum is divisible by 9.	729 → 18
10	Last digit is 0.	450
11	Difference of alternating digit sums is 0 or a multiple of 11.	121
12	Divisible by both 3 and 4.	144

Example: Divisibility by 9

Check 5832. Add the digits: \(5+8+3+2=18\). Since 18 is divisible by 9, 5832 is divisible by 9.

4. Factors and Multiples

Factors

A factor divides a number exactly. Factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.

Multiples

Multiples are found by multiplying a number by 1, 2, 3, 4 and so on. Multiples of 6 are 6, 12, 18, 24, 30...

Prime Factorisation

Write a number as a product of prime numbers.

\[72=2^3\times3^2\]

Number of Factors

If \(N=p^a q^b r^c\), then the number of positive factors is:

\[(a+1)(b+1)(c+1)\]

For \(72=2^3\times3^2\): factors = \((3+1)(2+1)=12\).

5. HCF and LCM

HCF means Highest Common Factor. LCM means Lowest Common Multiple.

\[\operatorname{HCF}(a,b)\times\operatorname{LCM}(a,b)=a\times b\]

Example: 24 and 36

\(24=2^3\times3\), \(36=2^2\times3^2\).

HCF = \(2^2\times3=12\).

LCM = \(2^3\times3^2=72\).

Euclidean Method

\[a=bq+r\]

Keep dividing until the remainder becomes 0. The last non-zero remainder is the HCF.

6. Remainders

\[\text{Dividend}=\text{Divisor}\times\text{Quotient}+\text{Remainder}\]\[0\le r

\(47=5\times9+2\). So the remainder is 2.

Same Remainder Idea

If two numbers leave the same remainder when divided by a number, their difference is exactly divisible by that number.

7. Unit Digit

For large powers, look only at the last digit and find its repeating cycle.

Base	Cycle
2	2,4,8,6
3	3,9,7,1
4	4,6
7	7,9,3,1
8	8,4,2,6
9	9,1

Example: Unit digit of \(2^{25}\). Cycle length = 4. Since \(25\mod4=1\), take the first cycle value. Answer = 2.

8. Fractions and Decimals

Proper Fraction

Numerator is smaller than denominator, for example \(\frac35\).

Improper Fraction

Numerator is greater than or equal to denominator, for example \(\frac75\).

Comparing Fractions

\[\frac ab>\frac cd\iff ad>bc\quad(b,d>0)\]

Recurring Decimals

\[0.\overline3=\frac13\]\[0.\overline{27}=\frac{27}{99}=\frac3{11}\]

9. Powers, Squares, Cubes and Roots

\[a^m\times a^n=a^{m+n}\]\[\frac{a^m}{a^n}=a^{m-n}\quad(a\ne0)\]\[(a^m)^n=a^{mn}\]\[(ab)^n=a^nb^n\]

Perfect squares: 1, 4, 9, 16, 25, 36...

Perfect cubes: 1, 8, 27, 64, 125...

10. SSC GD Solved Questions

Q1. Is 756 divisible by 6?

756 is divisible by 2 because its last digit is 6. Digit sum is \(7+5+6=18\), divisible by 3. So 756 is divisible by 6.

Q2. Find HCF of 48 and 72.

\(48=2^4\times3\), \(72=2^3\times3^2\). HCF = \(2^3\times3=24\).

Q3. Find LCM of 18 and 24.

\(18=2\times3^2\), \(24=2^3\times3\). LCM = \(2^3\times3^2=72\).

Q4. Find remainder when 125 is divided by 9.

\(125=9\times13+8\). Answer = 8.

Q5. Find unit digit of \(3^{17}\).

Cycle is 3, 9, 7, 1. \(17\mod4=1\). Answer = 3.

Q6. How many factors does 72 have?

\(72=2^3\times3^2\). Number of factors = \((3+1)(2+1)=12\).

Q7. Find the greatest number dividing 85 and 115 leaving the same remainder.

Difference = \(115-85=30\). Answer = 30.

Q8. Find the smallest number divisible by 12, 15 and 20.

LCM of 12, 15 and 20 is 60. Answer = 60.

Q9. Which is greater: \(\frac7{12}\) or \(\frac58\)?

\(7\times8=56\) and \(5\times12=60\). Therefore \(\frac58\) is greater.

Q10. Find the sum of first 20 natural numbers.

Use \(n(n+1)/2\). So \(20\times21/2=210\).

11. Practice Set

1. Find HCF of 54 and 90.
2. Find LCM of 16 and 24.
3. Is 4536 divisible by 9?
4. Find the unit digit of \(8^{15}\).
5. Find number of factors of 100.
6. Find remainder when 999 is divided by 8.
7. Find the smallest number divisible by 8, 12 and 15.
8. Compare \(\frac{11}{15}\) and \(\frac7{10}\).
9. Find unit digit of \(4^{35}\).
10. Find HCF × LCM for 18 and 30.

Answer Key

1. 18 • 2. 48 • 3. Yes • 4. 8 • 5. 9 • 6. 7 • 7. 120 • 8. \(\frac{11}{15}\) • 9. \(\frac7{10}\) • 10. 540

12. Quick Revision

\[\text{Dividend}=\text{Divisor}\times\text{Quotient}+\text{Remainder}\]\[HCF\times LCM=a\times b\]\[d(N)=(a+1)(b+1)(c+1)\]\[0\le r

Common mistakes:

• Do not confuse factors and multiples.
• Remainder is always smaller than divisor.
• For divisibility by 6, check both 2 and 3.
• For unit digit questions, check the cycle first.
• Cross multiply carefully while comparing fractions.

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