1400 = 40*35 = 2^3 * 5^2 * 7
So, the only missing odd factor is 5
LCM(35*5,40) = LCM(175,40) = 1400
So, the only missing odd factor is 5
LCM(35*5,40) = LCM(175,40) = 1400
Let's start with the prime factorization of 40:
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.
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.