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

To find an odd natural number x such that the least common multiple (LCM) of x and 40 is 1400, we can start by listing the prime factorization of 1400.

The prime factorization of 1400 is 2^3 × 5^2 × 7.

Since the LCM is the product of the highest power of each prime factor appearing in the factorizations of x and 40, we need to find a number x that includes the missing factors to make the LCM equal to 1400.

For x to have a LCM of 1400 with 40, it must include the highest power of 7, which is 7^1, and the highest power of 5, which is 5^2. Additionally, if x is an odd number, there should be no factor of 2 in x.

Now, let's consider the prime factorization of x.

x = 7^1 × 5^2 × p, where p is a prime factorization including no factor of 2.

To satisfy the condition of being an odd number, we need to choose a value for p that includes no factors of 2. It means p should be a prime factorization of an odd number.

For example, let's choose p = 3, which is an odd prime.

Therefore, x = 7^1 × 5^2 × 3 = 525.

Thus, an odd natural number x that satisfies the condition is 525.