# Find the value of odd natural number x if LCM (x,40)=1400.

## To find the value of the odd natural number x, we first need to determine the prime factorization of 1400 and analyze its factors.

Step 1: Prime factorization of 1400

To find the prime factorization of 1400, we divide it by the smallest prime number, which is 2.

1400 ÷ 2 = 700

Further, we divide 700 by 2 again:

700 ÷ 2 = 350

Once more for 350:

350 ÷ 2 = 175

We can see that we can't divide 175 by 2 any further, so we move on to the next prime number, which is 3. However, 175 is not divisible by 3.

Next, we try dividing 175 by 5:

175 ÷ 5 = 35

Now, we try dividing 35 by 5:

35 ÷ 5 = 7

At this point, we can see that we cannot divide 7 by any prime number, so the prime factorization of 1400 is:

2 × 2 × 2 × 5 × 5 × 7

Step 2: Analyzing the factorization and determining the odd natural number

From the prime factorization of 1400, we can see that it contains two factors of 2 (2 × 2) and no other prime factors of 2. Thus, the odd natural number x must not contain any factors of 2.

To find the LCM (Least Common Multiple) of x and 40, we need to consider their prime factorizations, considering only the primes that are not common to both x and 40.

Prime factorization of 40:

40 ÷ 2 = 20

20 ÷ 2 = 10

10 ÷ 2 = 5

The prime factorization of 40 is: 2 × 2 × 2 × 5

Comparing the prime factorizations of 1400 and 40, we can see that the common factor is 2 × 2 × 2 × 5 = 40.

To calculate the LCM, we multiply the common factors by the remaining unique factors:

LCM (x, 40) = (2 × 2 × 7) × (2 × 2 × 2 × 5) = 28 × 40 = 1120

Since LCM (x, 40) is given as 1400, and x shouldn't have any factors of 2, the value of x can be calculated by dividing 1400 by 40:

x = 1400 ÷ 40 = 35

Therefore, the value of the odd natural number x is 35.