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

## 1400 = 40*35 = 2^3 * 5^2 * 7

So, the only missing odd factor is 5
LCM(35*5,40) = LCM(175,40) = 1400

## To find the value of the odd natural number x, we need to find the prime factors of 40 and 1400 and then determine the highest power of each prime factor.

40 = 2^3 * 5

Now, let's find the prime factorization of 1400:
1400 = 2^3 * 5^2 * 7

To find the LCM (least common multiple) of x and 40, we need to find the highest power of each prime factor in the factorization of 1400.

In the factorization of 1400, we only have 2^3 and 5^2, which means the highest power of 7 should be 0.

Therefore, the LCM of x and 40 should have the prime factors of 2^3 and 5^2. To have a LCM of 1400, the highest power of 2 should be 3, and the highest power of 5 should be 2.

Thus, the prime factorization of x would be x = 2^3 * 5^2.

Now, let's find the value of x:
x = 2^3 * 5^2
x = 8 * 25
x = 200

So, the value of the odd natural number x, for which LCM(x, 40) = 1400, is 200.

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

Prime factorization of 1400:
1400 = 2^3 * 5^2 * 7

Prime factorization of 40:
40 = 2^3 * 5

Since the LCM includes all the prime factors from both numbers, we can see that the LCM of x and 40 should have the same prime factors and at least the same exponents.

Comparing the prime factorization of 1400 and the LCM of x and 40:
LCM (x, 40) = 2^3 * 5^2 * 7

Now, we can determine the values of x by matching the prime factors and their exponents.

The highest power of 2 in x can be equal to or greater than the one in 1400 (2^3). Let's assume it is equal, so x has a factor of 2^3.

The highest power of 5 in x can also be equal to or greater than the one in 1400 (5^2). Let's assume it is equal, so x has a factor of 5^2.

Since x is an odd natural number, it cannot have 2 as its factor more than 3 times. Therefore, x cannot have a factor of 2^3.

Hence, there is no odd natural number x that satisfies LCM (x, 40) = 1400.