## To find the value of the odd natural number x, given that the least common multiple (LCM) of x and 40 is 1400, you can follow these steps:

Step 1: Prime Factorization

First, express 40 and 1400 as products of their prime factors:

- 40 = 2^3 * 5^1

- 1400 = 2^3 * 5^2 * 7^1

Step 2: LCM Calculation

The LCM of two numbers can be found by taking the highest power of each prime factor that appears in either number. In this case, the LCM of x and 40 is 1400, so we can compare the exponents of the prime factors:

- For the prime factor 2:

The exponent of 2 in 1400 is 3, which is also the exponent in 40. Therefore, x must have an exponent of 3 for the prime factor 2.

- For the prime factor 5:

The exponent of 5 in 1400 is 2, while the exponent in 40 is 1. Therefore, x must have an exponent of 2 for the prime factor 5.

- For the prime factor 7:

The exponent of 7 in 1400 is 1, but there is no exponent of 7 in 40. Therefore, x does not have a factor of 7.

Step 3: Forming the Odd Natural Number x

Based on the prime factorization of x, we now have:

- The prime factor 2 appears with an exponent of 3.

- The prime factor 5 appears with an exponent of 2.

- There is no prime factor 7.

To form an odd natural number, we cannot include the prime factor 2 in x. Therefore, x would only consist of the prime factor 5 with an exponent of 2, which means x equals 5^2 = 25.

Hence, the value of the odd natural number x is 25.