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Number System & Problems on Numbers

Practice MCQs

The number system is a way of representing and expressing numerical values. It provides a systematic framework for counting, measuring, performing calculations, and expressing quantities.

Quantitative Aptitude Number System & Problems on Numbers Competitive Exams

Number System & Problems on Numbers is a foundational quantitative aptitude topic. It includes types of numbers, divisibility rules, factors, multiples, remainders, place value, face value, unit digit, number properties, and word problems based on numbers.

What is the Number System?

The number system is the method of representing and classifying numbers. In competitive exams, number system questions usually test your understanding of natural numbers, whole numbers, integers, rational numbers, prime numbers, composite numbers, odd-even numbers, divisibility, and remainders.

Problems on Numbers are word problems where unknown numbers are represented using variables and solved using equations. These questions often involve sums, differences, products, digits, ratios, and conditions.

Quick idea: In number problems, convert the statement into an equation. In number system questions, identify the property being tested.
Concept Meaning Example
Natural Numbers Counting numbers starting from 1 1, 2, 3, 4, ...
Whole Numbers Natural numbers including zero 0, 1, 2, 3, ...
Integers Positive, negative numbers and zero ..., -2, -1, 0, 1, 2, ...
Prime Numbers Numbers having exactly two factors 2, 3, 5, 7, 11
Composite Numbers Numbers having more than two factors 4, 6, 8, 9, 10

“Most number problems become simple when the unknown number is represented clearly.”

Aptitude Tip
Key points
  • Zero is a whole number but not a natural number.
  • 2 is the only even prime number.
  • 1 is neither prime nor composite.
  • Every even number is divisible by 2.
  • Place value depends on position of digit.
  • Use variables for unknown number problems.
numbers digits divisibility remainders

Visual Understanding

The following diagrams show how numbers are classified and how place value works.

Number Classification
Real Numbers Rational Numbers Integers ..., -2, -1, 0, 1, 2, ... Whole Numbers: 0, 1, 2, 3, ... Natural numbers: 1, 2, 3, 4, ...

Natural numbers are part of whole numbers, and whole numbers are part of integers.

Place Value of Digits
4 7 3 5 2 Ten
Thousands Thousands Hundreds Tens Ones Number = 47,352
\[ 47352 = 40000 + 7000 + 300 + 50 + 2 \]

Place value changes according to the position of the digit.

Important Rules and Formulas

Even Number
\[ n = 2k \]

Any number divisible by \(2\) is even.

Odd Number
\[ n = 2k+1 \]

Odd numbers are not divisible by \(2\).

Two-Digit Number
\[ 10x+y \]

Here \(x\) is tens digit and \(y\) is units digit.

Three-Digit Number
\[ 100x+10y+z \]

Here \(x\), \(y\), and \(z\) are digits.

Consecutive Numbers
\[ x,\ x+1,\ x+2 \]

Used when numbers are one after another.

Consecutive Even Numbers
\[ x,\ x+2,\ x+4 \]

Difference between consecutive even numbers is \(2\).

Consecutive Odd Numbers
\[ x,\ x+2,\ x+4 \]

Difference between consecutive odd numbers is also \(2\).

Division Algorithm
\[ N = dq + r \]

Here \(N\) is number, \(d\) divisor, \(q\) quotient, \(r\) remainder.

Rule: In digit-based questions, always express the number using place value, such as \(10x+y\) for a two-digit number.

Important Divisibility Rules

Divisible By Rule Example
2 Last digit is even. 246 is divisible by 2.
3 Sum of digits is divisible by 3. 123: \(1+2+3=6\), divisible by 3.
4 Last two digits are divisible by 4. 1316: last two digits 16, divisible by 4.
5 Last digit is 0 or 5. 235 is divisible by 5.
6 Number is divisible by both 2 and 3. 234 is divisible by 6.
8 Last three digits are divisible by 8. 3120: last three digits 120, divisible by 8.
9 Sum of digits is divisible by 9. 729: \(7+2+9=18\), divisible by 9.
10 Last digit is 0. 450 is divisible by 10.
11 Difference between sum of alternate digits is 0 or divisible by 11. 121: \((1+1)-2=0\), divisible by 11.
Tip: Divisibility rules help solve number system, simplification, HCF-LCM, and remainder questions quickly.

Common Types of Questions

Number Classification

Identify type of number.

  • Natural numbers
  • Whole numbers
  • Integers
  • Prime and composite
Digit-Based Problems

Questions based on digits and reversed numbers.

  • Two-digit numbers
  • Digit sum
  • Reversed number
  • Place value
Consecutive Numbers

Numbers appearing one after another.

  • Consecutive integers
  • Consecutive even numbers
  • Consecutive odd numbers
  • Sum-based equations
Remainder Problems

Questions based on division and remainder.

  • Dividend
  • Divisor
  • Quotient
  • Remainder
Exam approach: First identify whether the problem is about number type, divisibility, digit relation, consecutive numbers, or remainder.
Method Bank
Two-Digit Number

If tens digit is \(x\) and units digit is \(y\):

\[ N = 10x+y \]
Reversed Number

If original number is \(10x+y\):

\[ R = 10y+x \]
Sum of Consecutive Numbers

For \(x,\ x+1,\ x+2\):

\[ \text{Sum} = 3x+3 \]
Dividend Relation

For division problems:

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

Tip: For word problems, define the unknown number as \(x\), then translate the sentence into an equation.

Prime and Composite Numbers
Prime 2, 3, 5 7, 11, 13 Exactly two factors Composite 4, 6, 8 9, 10, 12 More than two factors Note: 1 is neither prime nor composite
Prime numbers have exactly two factors. Composite numbers have more than two factors.

Step-by-Step Solving Method

Step Action Example
Step 1 Read the question and identify the number type. Two consecutive numbers, digit problem, remainder problem, etc.
Step 2 Represent unknown number using a variable. Let the number be \(x\).
Step 3 Translate the sentence into an equation. Twice a number plus 5 is 21: \(2x+5=21\).
Step 4 Solve the equation carefully. \(2x=16\), so \(x=8\).
Step 5 Check the answer with the original condition. Twice 8 plus 5 is 21.
Important: In digit problems, do not treat digits as ordinary numbers. Use place value, such as \(10x+y\) for a two-digit number.

Solved Examples

Question Method Answer
Find the number if twice the number plus 5 is 21. Let the number be \(x\).
\[ 2x+5=21 \]
\[ 2x=16 \]
\[ x=8 \]
8
The sum of three consecutive numbers is 48. Find the numbers. Let the numbers be \(x,\ x+1,\ x+2\).
\[ x+(x+1)+(x+2)=48 \]
\[ 3x+3=48 \]
\[ x=15 \]
15, 16, 17
The sum of two consecutive even numbers is 46. Find the numbers. Let the numbers be \(x\) and \(x+2\).
\[ x+(x+2)=46 \]
\[ 2x+2=46 \]
\[ x=22 \]
22 and 24
A two-digit number has tens digit 4 and units digit 7. Find the number. Tens digit \(=4\), units digit \(=7\).
\[ N = 10x+y \]
\[ N = 10(4)+7 = 47 \]
47
A two-digit number has digits whose sum is 9. If the number is 27 more than the reversed number, find the number. Let tens digit be \(x\), units digit be \(y\).
\[ x+y=9 \]
Original number \(=10x+y\), reversed number \(=10y+x\).
\[ (10x+y)-(10y+x)=27 \]
\[ 9x-9y=27 \]
\[ x-y=3 \]
Solving \(x+y=9\) and \(x-y=3\), we get \(x=6,\ y=3\). 		63
When a number is divided by 7, quotient is 8 and remainder is 5. Find the number.
\[ N = dq+r \]
\[ N = 7\times8+5 \]
\[ N = 61 \]
61
Check whether 729 is divisible by 9. Add the digits:
\[ 7+2+9=18 \]
Since \(18\) is divisible by \(9\), \(729\) is divisible by \(9\). 		Yes
Find the smallest prime number. Prime numbers have exactly two factors. The number \(2\) has factors \(1\) and \(2\). It is also the only even prime number. 2

Note: Most number word problems are solved by forming equations from the given statements.

Common Traps and Shortcuts

Common Traps
  • Treating 1 as a prime number.
  • Forgetting that 0 is a whole number.
  • Confusing place value and face value.
  • Writing a two-digit number as \(x+y\) instead of \(10x+y\).
  • Using wrong divisibility rule.
  • Forgetting that remainder must be less than divisor.
Useful Shortcuts
  • 2 is the only even prime number.
  • For divisibility by 3 or 9, use digit sum.
  • For divisibility by 4, check last two digits.
  • For divisibility by 8, check last three digits.
  • For digit reversal, use \(10x+y\) and \(10y+x\).
  • For consecutive numbers, use \(x,\ x+1,\ x+2\).
Exam approach: Identify the number property first. Then use the relevant rule, equation, or place value representation.

Practice

A) Multiple Choice Questions
  1. Which of the following is the only even prime number?
    1 2 4 6
  2. Which number is neither prime nor composite?
    0 1 2 3
  3. Find the number if \(3x+4=25\).
    5 6 7 8
  4. A number has tens digit 5 and units digit 8. The number is:
    13 45 58 85
  5. Which of the following is divisible by 9?
    124 235 459 572
B) Solve the Higher-Order Problems
  1. The sum of three consecutive integers is 72. Find the integers. (Hint: Use \(x,\ x+1,\ x+2\).)
  2. The sum of two consecutive odd numbers is 56. Find the numbers. (Hint: Use \(x\) and \(x+2\).)
  3. A two-digit number has digits whose sum is 12. If the number is 18 more than the reversed number, find the number. (Hint: Use \(10x+y\) and \(10y+x\).)
  4. When a number is divided by 9, quotient is 12 and remainder is 4. Find the number. (Hint: Use \(N=dq+r\).)
  5. Check whether 3726 is divisible by 3 and 9. (Hint: Use sum of digits.)
C) Match the Concept with the Correct Meaning
Concept Correct Meaning
Natural Numbers Counting numbers starting from 1
Whole Numbers Natural numbers including zero
Prime Number Number with exactly two factors
Composite Number Number with more than two factors
Place Value Value of digit according to its position
Remainder Amount left after division
Aptitude Reminder

Number system questions require clear understanding of number types, divisibility rules, place value, and remainders. Problems on numbers usually require converting a statement into an equation.

Task: Create five questions using prime numbers, divisibility rules, digit reversal, consecutive numbers, and remainder formula.

Show Suggested Answers
Multiple Choice
  1. 2
    \(2\) is the only even prime number.
  2. 1
    \(1\) is neither prime nor composite.
  3. 7
    \[ 3x+4=25 \]
    \[ 3x=21 \]
    \[ x=7 \]
  4. 58
    Tens digit \(=5\), units digit \(=8\).
    \[ N=10(5)+8=58 \]
  5. 459
    \[ 4+5+9=18 \]
    Since \(18\) is divisible by \(9\), \(459\) is divisible by \(9\).
Higher-Order Problems
  1. Let the integers be \(x,\ x+1,\ x+2\).
    \[ x+(x+1)+(x+2)=72 \]
    \[ 3x+3=72 \]
    \[ x=23 \]
    Answer = 23, 24, 25.
  2. Let the odd numbers be \(x\) and \(x+2\).
    \[ x+(x+2)=56 \]
    \[ 2x+2=56 \]
    \[ x=27 \]
    Answer = 27 and 29.
  3. Let tens digit be \(x\), units digit be \(y\).
    \[ x+y=12 \]
    Original number \(=10x+y\), reversed number \(=10y+x\).
    \[ (10x+y)-(10y+x)=18 \]
    \[ 9x-9y=18 \]
    \[ x-y=2 \]
    Solving \(x+y=12\) and \(x-y=2\), we get \(x=7,\ y=5\).
    Answer = 75.
  4. Divisor \(=9\), quotient \(=12\), remainder \(=4\).
    \[ N=dq+r \]
    \[ N=9\times12+4=112 \]
    Answer = 112.
  5. For \(3726\):
    \[ 3+7+2+6=18 \]
    Since \(18\) is divisible by \(3\) and \(9\), \(3726\) is divisible by both \(3\) and \(9\).
    Answer = Yes, divisible by both 3 and 9.
Concept Matching
  1. Natural Numbers → Counting numbers starting from 1
  2. Whole Numbers → Natural numbers including zero
  3. Prime Number → Number with exactly two factors
  4. Composite Number → Number with more than two factors
  5. Place Value → Value of digit according to its position
  6. Remainder → Amount left after division
Clue Explanation

For number problems, convert the given statement into a mathematical equation. For digit problems, always use place value representation.

Exam tips
  • Remember that 1 is neither prime nor composite.
  • Use digit sum for divisibility by 3 and 9.
  • Use \(10x+y\) for two-digit numbers.
  • Use \(N=dq+r\) for remainder problems.
  • For consecutive numbers, define them carefully.
  • Always verify final answer with original condition.
Practice MCQs