# find the value of odd natural number x if lcm(x,40)=1400

## To find the value of the odd natural number x, we will use the concept of the least common multiple (LCM).

The LCM of two numbers is the smallest positive multiple that both numbers share. In this case, we are given that the LCM of x and 40 is equal to 1400.

To find the LCM of x and 40, we need to find the prime factorization of both numbers.

Let's start with the number 40. The prime factorization of 40 is 2^3 * 5^1.

Now, let's represent x as the product of its prime factors raised to their respective powers. Since x is an odd natural number, it cannot have any power of 2. Thus, the prime factorization of x will only contain odd prime factors.

So, the prime factorization of x is x = p^a * q^b * r^c * ... where p, q, r, etc. are odd prime numbers.

Now, we can find the LCM of x and 40 by taking the highest power of each prime factor that appears in x and 40.

For the prime factor 2, the highest power it appears in 40 is 2^3. However, since x is odd, it does not have any power of 2. Therefore, the power of 2 in the LCM of x and 40 will be 2^3.

For the prime factor 5, it appears with power 1 in 40. We don't know the power of 5 in x, but we can deduce that it must be 1 since x is odd. Therefore, the power of 5 in the LCM of x and 40 will be 5^1.

For any other prime factors of x (represented as p, q, r, etc.), we don't know their powers, but we can assume them to be 0 since those prime factors are not present in x (odd numbers only have odd prime factors).

Now, we can write the LCM of x and 40 in terms of their prime factorization:

LCM(x, 40) = 2^3 * 5^1 * p^0 * q^0 * r^0 * ...

Since we are given that the LCM(x, 40) is equal to 1400, we can equate the prime factorization of the LCM with the prime factorization of 1400, which is 2^3 * 5^2 * 7^1.

Therefore, we can set up the following equation:

2^3 * 5^1 * p^0 * q^0 * r^0 * ... = 2^3 * 5^2 * 7^1

From this equation, we can conclude that p, q, r, etc. are not present in the prime factorization of 1400, so we can assume their powers to be 0.

Now, let's solve this equation:

2^3 * 5^1 * 1 * 1 * 1 * ... = 2^3 * 5^2 * 7^1

Canceling out the common factors (2^3 and 5^1) from both sides of the equation, we get:

1 * 1 * 1 * ... = 5^1 * 7^1

Simplifying further, we have:

1 = 5 * 7

As we can see, this equation is not true. There is no value of x that satisfies the given condition LCM(x, 40) = 1400 when x is an odd natural number.