Two positive integers are in the ratio 3 : 4. If their LCM is 192, what is the difference between the squares of the numbers?
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Data given
Let the two numbers be 3x and 4x.
Let us find the LCM of two numbers 3x and 4x by the prime factorization method.
Method: The LCM of the two numbers is the product of all the different prime factors of both the numbers in their respective highest powers.
Prime factorize 3x: 3 * x
Prime factorize 4x: 2^2 * x
So, the LCM of 3x and 4x = 3 * 2^2 * x = 12x
Equating 12x to the LCM of the two numbers we get, 12x = 192
Or x = 16.
So, the two numbers are 3*16 and 4*16
i.e., the numbers are 48 and 64
The numbers are 48 and 64.
The difference between their squares is 64^{2} - 48^{2} = (64 + 48)(64 - 48)
= 112 * 16 = 1792
The difference between the squares of the numbers is 1792.