find the lcm of 140 and 168 using formula whose hcf is 28 please answer me
Answers
Answer:
LCM = 112
Step-by-step explanation:
Let the two numbers be x and y. We can write
x = 28 * 7^a * b
y = 28 * 2^c * d
where b and d aren’t divisible by 2 nor 7, since 2 and 7 are the prime factors of 28 and if both numbers have more powers of one of those two factors, then they have a higher HCF.
x clearly can’t have another power of 7 since the sum of the two numbers would then exceed 140 since both numbers are positive. In fact, x can at best only add a power of 3, so x = 28 or 28 * 3 = 84, since 28 * 5 = 140 which would make the sum exceed 140.
y can at best have two more powers of 2, so y = 28, 56 or 112 (28 * 3 is covered by x, and they can’t both be 84 at the same time since the sum would then exceed 140).
By inspection, there are two possible solutions: x = 28 and y = 112, and x = 84 and y = 56.
The first solution has LCM(x, y) = 112 since y just adds prime powers that are already present in x. Thus, the two numbers are 56 and 84.
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