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CAT 2025 Lesson : Number Theory - Arithmetic Operations

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2.4 Arithmetic operations

While performing mathematical calculations, we must operate in order.

Eg., 3+42=3+23 + \dfrac{4}{2} = 3 + 2 and not 3+42\dfrac{3 + 4}{2}

Eg., 3×5−2=15−2=133 \times 5 - 2 = 15 - 2 = 13 and not 3×3=93 \times 3 = 9.

The rules of which mathematical operator comes first are denoted by BODMAS

2.4.1 BODMAS

BODMAS is an acronym, where each letter (as stated below) represents an arithmetic operation.

B = Brackets
O = Orders (i.e. Powers and Square Roots, Cube Roots, etc.)
D = Division
M = Multiplication
A = Addition
S = Subtraction

BODMAS represents the order in which arithmetic operations are to be performed in an expression, from left to right.

In other words, operations involving Brackets are completed first, followed by those involving Orders (Powers), Division, Multiplication, Addition and finally Subtraction.

We should always follow this BODMAS order while solving questions. If not our answers might go wrong.

The order in which the operations within different types of brackets are to be done is as follows

1) 123‾\overline{\color{white} 123} called line bracket. Eg, 4×5+3‾=4×8=324 \times \overline{5 + 3} = 4 \times 8 = 32
2) () called parenthesis or common bracket
3) {} called curly bracket
4) [] called rectangular bracket

Example 8

8+3×(4×5−6‾)×{15−12}=?8 + 3 \times (4 \times \overline{5 - 6}) \times \{15 - 12 \}= ?

Solution

8+3×(4×5−6‾)×{15−12}8 + 3 \times (4 \times \overline{5 - 6}) \times \{15 - 12 \}
== 8+3×(4×−1)×38 + 3 \times (4 \times -1) \times 3
== 8+3×−4×38 + 3 \times -4 \times 3
== 8−368-36
== −28-28
Answer: −28-28


2.4.2 Basics of Remainders

When a number say nn (also called dividend) is divided by divisor (dd), it leaves a quotient (qq) and remainder (rr).



So, nn can be expressed as n=dq+rn = dq + r

For example, when 3737 is divided by 99, we get a quotient of 44 and remainder of 11.

37=9×4+137 = 9 \times 4 + 1


Example 9

14871487, when divided by xx, leaves a remainder of 55. What is the largest three-digit number that xx can be?

Solution

The dividend is 14871487 and remainder is 55. xx is the divisor. Let the quotient be qq.

1487=xq+51487 = xq + 5

⇒ xq=1482xq = 1482

⇒ x=1482qx = \dfrac{1482}{q}

Value of xx is highest when qq is lowest.

When q=1q = 1, we get a 44-digit number.

When q=2q = 2, x=14822=741x = \dfrac{1482}{2} = 741, which is the largest possible 33-digit number.

Answer: 741741

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