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Number Theory

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CAT 2025 Lesson : Number Theory - Other Numerical Operations

3. Other Numerical Operations

3.1 Square Root

Calculating square roots, cube roots, etc. by applying the prime factorisation method is detailed in the Surds & Indices lesson. This section will cover the Long division method for square roots only.

The steps to be followed can be best explained with an example. Let's find the square root of

55696

, i.e.

\sqrt{55696}

Step 1: Starting from the right end, insert commas after every 2 digits.

55696

is split into three parts and written as

\bm{5 , 56 , 96}

Step 2: Write the largest number whose square is less than the first part as the quotient (on the top) and divisor (on the left).

Step 3: After subtracting this square from the dividend, bring down the next pair of digits.

Step 4: Add the units digit of the divisor to itself. Find the largest possible digit which can then be added to the divisor such that the product of the digit and the divisor is less than the first part of the dividend brought down.

Step 5: Add this digit to the quotient. Subtract the product from the dividend and bring down the next pair of digits.

Steps 4 and 5 continue till we get a remainder of zero or the decimal at which we wish to approximate.

In the above example,

\sqrt{55696} = 236

.

As shown in the example below, in case of irrational square roots, we bring down zeroes in pairs after the decimal point.

Example 10

Approximate

\sqrt{70}

to two decimal places.

Solution

We stop at three decimal places so that we can approximate.

\sqrt{70} = 8.366.. \sim 8.37

Answer:

8.37

3.2 Modulus

The modulus function of a number provides the non-negative value of the number. To define it as a function,

|x| = x

x \ge 0

\space \space \space \space \space \space= -x

x \lt 0

Example 11

How many integral values of

x

satisfy

|x - 2 | \lt 4

Solution

\bm{x \gt 2}

, then

x - 2 \lt 4

⇒

x \lt 6

Possible values of

x

are

3

4

and

5

.

If

\bm{x \lt 2}

, then

-(x - 2) \lt 4

⇒

-x \lt 2

x \gt -2

Possible values of

x

are

-1

0

and

1

.

If

\bm{x = 2}

, then condition is satisfied.

\therefore x = 2

is also a possible value.

\therefore

Total of

7

possible values.

Answer:

7

3.3 Factorial

A Factorial is represented by an exclamation mark or '

!

'. '

\bm{n!}

' is read as '

\bm{n}

factorial'.

\bm{n!}

is the product of all natural numbers less than or equal to

\bm{n}

\therefore n! = 1 \times 2.... \times n

Exception: Although 0 is not a natural number,

\bm{0! = 1}

To enhance your speed, you should memorise the values of the following commonly used factorials.

$0! = 1$	$1! = 1$	$2! = 2$	$3! = 6$	$4! = 24$
$5! = 120$	$6! = 720$	$7! = 5040$	$8! = 40320$

Note the following properties with factorials.

1)

n! \times (n + 1) = (n + 1)!

(n + 1)! - n! = n \times n!

Example 12

Which of the following equals

10!

?

(1)

9! \times 9

(2)

\dfrac{11!}{10}

(3)

8! \times 100

(4)

8! \times 90

Solution

8! \times 90 = 8! \times 9 \times 10 = 10!

Answer: (4)

8! \times 90

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