# Find an odd natural number x such that LCM (x, 40) = 1400

## For 40 the prime factorization is:

2³ ∙ 5¹

For 1400 the prime factorization is:

2³ ∙ 5² ∙ 7¹

Since x is an odd number, the factor 2³ must be discarded.

The number x can be written as:

x = 5ᵐ ∙ 7ⁿ

where

m ≤ 2

because in factorization the largest exponent of the number 5 is two

n ≤ 1

because in factorization the largest exponent of the number 7 is one

By testing these values you get:

For m = 0 , n = 0

x = 5⁰ ∙ 7⁰ = 1 ∙ 1 = 1

LCM of 1 and 40 is 40

For m = 0 , n = 1

x = 5⁰ ∙ 7¹ = 1 ∙ 7 = 7

LCM of 7 and 40 is 280

For m = 1 , n = 0

x = 5¹ ∙ 7⁰ = 5 ∙ 1 = 5

LCM of 5 and 40 is 40

For m = 1 , n = 1

x = 5¹ ∙ 7¹ = 5 ∙ 7 = 35

LCM of 35 and 40 is 280

For m = 2 , n = 0

x = 5² ∙ 7⁰ = 25 ∙ 1 = 25

LCM of 25 and 40 is 200

For m = 2 , n = 1

x = 5² ∙ 7¹ = 25 ∙ 7 = 175

LCM of 175 and 40 is 1400

So x = 175

## To find an odd natural number x such that LCM(x, 40) = 1400, we need to determine the prime factorization of 1400 and then find a number x that includes some or all of the prime factors of 1400.

Step 1: Prime Factorization of 1400
To find the prime factorization of 1400, we can repeatedly divide it by prime numbers until we reach the point where the quotient is no longer divisible by any prime number. Let's start:

1400 ÷ 2 = 700
700 ÷ 2 = 350
350 ÷ 2 = 175
175 ÷ 5 = 35
35 ÷ 5 = 7

So, the prime factorization of 1400 is 2 × 2 × 2 × 5 × 5 × 7, which can be represented as 2^3 × 5^2 × 7.

Step 2: Considering Odd Factors
Since we want an odd number, we need to exclude all the factors of 2 from the prime factorization of 1400. Therefore, we remove the powers of 2 and keep only the remaining factors.

The reduced prime factorization of 1400 without any powers of 2 becomes 5^2 × 7.

Step 3: Find the Odd Number
To find an odd number x, we can assign any value to the powers of 5 and 7, as long as they are greater than or equal to 1. However, we want the least odd number possible, so we choose the smallest values for the powers of 5 and 7.

Let's set the power of 5 to 1 and the power of 7 to 1:
x = 5^1 × 7^1
x = 5 × 7
x = 35

Therefore, the odd natural number x that satisfies LCM(x, 40) = 1400 is x = 35.

## odd means = 2n +1

then 1400 = 40x = 40( 2n + 1)
then 1400/40 = x/2n + 1
35 = x/2n +1
x = 35(2n + 1)
test n with small number start from 0
n = 0 - - - - - - - - x = 35 it does't give 1400 when we take LCM
n = 1 -------- ------ x = 105 it satisfy all the property
n = 2 ---------------x = 175 " " " " " " "