SSC CGL Maths · Number System

Number Types, Primes and Squares

Every number belongs to a family (natural, whole, integer, rational, irrational, real), and a few of those families have quick tests you can run in your head.

All Number System notes

Why this matters

Fourteen questions in the bank, all EASY, and twelve of the thirteen questions from the 2025 shifts sit on this page. Each one is a single fact or a single bracket between two squares, so it should cost you far less than the 36 seconds Tier I gives a question.

Concept 1 of 4: The families of numbers and how they nest

Think of the families as boxes inside boxes. The counting numbers sit inside the whole numbers, which sit inside the integers, which sit inside the rationals. The irrationals sit beside the rationals, and the two together fill the real number line.

Definition

Natural numbers start at 1. Whole numbers are 0 together with the natural numbers.
Integers add the negatives: $\ldots, -2, -1, 0, 1, 2, \ldots$
A rational number can be written as $\frac{p}{q}$ with $p, q$ integers and $q \ne 0$ . Its decimal either stops or repeats.
An irrational number cannot be written that way. Its decimal never stops and never repeats: $\sqrt{2}$ , $\sqrt{7}$ , $\pi$ .
The square root of a natural number is an integer when the number is a perfect square, and irrational otherwise.
Rational plus irrational is always irrational. A non-zero rational times an irrational is irrational, but $0 \times \sqrt{3} = 0$ is rational, so 'rational times irrational' is only sometimes irrational.
Even numbers are the multiples of 2; odd numbers leave remainder 1 on division by 2. Two consecutive even numbers, or two consecutive odd numbers, differ by 2, so write them as $n$ and $n + 2$ .

Family	What it contains	Examples
Natural numbers	the counting numbers, starting at 1	1, 7, 250
Whole numbers	0 and every natural number	0, 1, 2, 15 0 is whole but not natural.
Integers	the whole numbers and their negatives	$-4$ , 0, 7
Rational numbers	every $\frac{p}{q}$ with $p, q$ integers and $q \ne 0$	$\frac{3}{4}$ , $-2$ , 1.5, $0.333\ldots$
Irrational numbers	real numbers that are not $\frac{p}{q}$	$\sqrt{2}$ , $\sqrt{7}$ , $\pi$
Real numbers	all rational and all irrational numbers	every point on the number line
Even numbers	integers that are multiples of 2	$-6$ , 0, 86, 88
Odd numbers	integers that leave remainder 1 on division by 2	$-7$ , 3, 15

Natural numbers sit inside the whole numbers, whole numbers inside the integers, integers inside the rationals; the rationals and the irrationals together make the real numbers.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 21 Sep 2025, Shift 2 · Q53Easy

Example 1 · Number System · Number Types, Primes and Squares

Which of the following is a non-zero number that belongs to every set: Whole, Integer, Rational, and Real?

Calling 0 a natural number

Natural numbers start at 1. Zero is a whole number, an integer and a rational number, but not a natural number.

Saying rational times irrational is always irrational

It is irrational only when the rational number is not zero. Since

0 \times \sqrt{3} = 0

, the honest answer is 'sometimes rational, sometimes irrational'.

Taking a repeating decimal for an irrational number

0.333\ldots = \frac{1}{3}

is rational. A decimal is irrational only when it never stops and never settles into a repeating block.

Drill 6 more on the families of numbers and how they nest

Concept 2 of 4: When a fraction has a terminating decimal

A decimal that stops is a fraction over 10, 100, 1000 and so on. Those numbers are built only from 2s and 5s, so a fraction can stop only when its denominator, after cancelling, is built only from 2s and 5s as well.

Definition

First write the fraction in lowest terms.
It has a terminating decimal exactly when the denominator has no prime factor other than 2 and 5.
Any other prime in the denominator (3, 7, 11, ...) gives a recurring decimal. Both kinds are rational.
With denominator $2^a \times 5^b$ , the decimal has as many places as the larger of $a$ and $b$ .
Fast route: the denominators 2, 4, 5, 8, 10, 16, 20, 25, 40 and 50 all terminate; 3, 6, 7, 9, 11, 12 and 15 never do (in lowest terms).

Terminating decimal test

\frac{p}{q} \ (\text{lowest terms}) \text{ terminates exactly when } q = 2^a \times 5^b

qthe denominator after cancelling
a, bwhole numbers, either may be 0

Worked example

Which of

\frac{7}{20}

\frac{5}{12}

and

\frac{9}{14}

has a terminating decimal? Write that one as a decimal.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 22 Sep 2025, Shift 1 · Q52Easy

Example 2 · Number System · Number Types, Primes and Squares

Which of the following numbers has a terminating decimal representation?

Judging the denominator before cancelling

\frac{6}{15}

looks recurring because 15 has a factor 3, but it cancels to

\frac{2}{5} = 0.4

. Always reduce to lowest terms first.

Drill 1 more on when a fraction has a terminating decimal

Concept 3 of 4: Prime numbers and the square-root test

If a number splits as a product of two factors, one of them is at most its square root. So to see whether a number is prime you only need to try the primes up to its square root, not every number below it.

Definition

A prime has exactly two factors: 1 and itself. 1 is not prime (it has one factor), and 2 is the only even prime.
To test $n$ , divide by the primes $2, 3, 5, 7, 11, 13, \ldots$ up to $\sqrt{n}$ . If none divides it, $n$ is prime.
The 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, 97.
Look-alikes that are NOT prime: $51 = 3 \times 17$ , $57 = 3 \times 19$ , $87 = 3 \times 29$ , $91 = 7 \times 13$ .
For a list of primes in a range, cross out evens, then multiples of 3 (digit sum), 5 (last digit) and 7; what survives below 121 is prime.

Worked example

Is 247 a prime number?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2023 · Tier I, 27 Jul 2023, Shift 3 · Q72Easy

Example 3 · Number System · Number Types, Primes and Squares

The average of all prime numbers between 32 and 69 is:

Counting 1 as a prime

A prime has exactly two factors. The number 1 has only one, so it is neither prime nor composite.

Letting 51, 57 or 91 through as prime

They look prime but are not:

51 = 3 \times 17

57 = 3 \times 19

91 = 7 \times 13

. Run the digit-sum check for 3 and try 7 before you accept a two-digit number.

Concept 4 of 4: Perfect squares near a given number

Perfect squares are fence posts along the number line: 1600, then 1681, then 1764. Any number sits between two posts, and the question is only ever which post is nearest, or how far away it is.

Definition

A perfect square is $k^2$ for a whole number $k$ . Its square root is an integer.
Bracket the number: find $k$ with $k^2 \le N < (k + 1)^2$ . Then the least number to add is $(k + 1)^2 - N$ , and the least to subtract is $N - k^2$ .
The least $n$ -digit square is the first square at or above $10^{n-1}$ : estimate the root, then step up by 1.
A square ends in 0, 1, 4, 5, 6 or 9, never in 2, 3, 7 or 8. Use this to strike options; it cannot prove a number is a square.
Fast route for a square root: the last digit narrows the root's last digit (a square ending in 9 has a root ending in 3 or 7), and the tens fix the rest. For 5329: $70^2 = 4900$ and $80^2 = 6400$ , so the root is 73 or 77, and $75^2 = 5625$ is too big, so 73.
Square a number near a round one with $(a + b)^2 = a^2 + 2ab + b^2$ : $73^2 = 4900 + 420 + 9 = 5329$ .

Bracketing between squares

k^2 \le N < (k + 1)^2: \ \text{add } (k + 1)^2 - N \text{ or subtract } N - k^2 \qquad (a + b)^2 = a^2 + 2ab + b^2

Nthe given number
kthe whole number whose square is just at or below N

Worked example

What is the least number that must be subtracted from 3000 to make it a perfect square? What is the least number that must be added instead?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 21 Sep 2025, Shift 1 · Q51Easy

Example 4 · Number System · Number Types, Primes and Squares

What is the smallest number that must be added to 2378 to make it a perfect square?

Going to the wrong square

The least number to add takes you UP to the next square; the least number to subtract takes you DOWN to the square below. Bracket the number first so you know both.

Trusting the last digit as proof

A number ending in 1, 4, 5, 6 or 9 may still not be a square (26 is not). The last digit can only strike options out; confirm a square by bracketing or squaring.

Drill 3 more on perfect squares near a given number

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

When a fraction has a terminating decimal
Terminating decimal test
$\frac{p}{q} \ (\text{lowest terms}) \text{ terminates exactly when } q = 2^a \times 5^b$
Perfect squares near a given number
Bracketing between squares
$k^2 \le N < (k + 1)^2: \ \text{add } (k + 1)^2 - N \text{ or subtract } N - k^2 \qquad (a + b)^2 = a^2 + 2ab + b^2$

Reference tables (1)

The families of numbers and how they nest8 rows

Family	What it contains	Examples
Natural numbers	the counting numbers, starting at 1	1, 7, 250
Whole numbers	0 and every natural number	0, 1, 2, 15 0 is whole but not natural.
Integers	the whole numbers and their negatives	$-4$ , 0, 7
Rational numbers	every $\frac{p}{q}$ with $p, q$ integers and $q \ne 0$	$\frac{3}{4}$ , $-2$ , 1.5, $0.333\ldots$
Irrational numbers	real numbers that are not $\frac{p}{q}$	$\sqrt{2}$ , $\sqrt{7}$ , $\pi$
Real numbers	all rational and all irrational numbers	every point on the number line
Even numbers	integers that are multiples of 2	$-6$ , 0, 86, 88
Odd numbers	integers that leave remainder 1 on division by 2	$-7$ , 3, 15

Natural numbers sit inside the whole numbers, whole numbers inside the integers, integers inside the rationals; the rationals and the irrationals together make the real numbers.

Watch out for (8)

Calling 0 a natural number
→ The families of numbers and how they nest
Saying rational times irrational is always irrational
→ The families of numbers and how they nest
Taking a repeating decimal for an irrational number
→ The families of numbers and how they nest
Judging the denominator before cancelling
→ When a fraction has a terminating decimal
Counting 1 as a prime
→ Prime numbers and the square-root test
Letting 51, 57 or 91 through as prime
→ Prime numbers and the square-root test
Going to the wrong square
→ Perfect squares near a given number
Trusting the last digit as proof
→ Perfect squares near a given number

Test yourself on Number System

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