The least common multiple of two numbers is 60, and one number is 7 less than the other number. What are the numbers? Justify your answer.
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The least common multipul of two numbers is 60 and one of the numbers is 7 less than the other number.what are the numbers?justify your answer
To find the numbers, we need to understand the concept of the least common multiple (LCM) and how it relates to the given information.
The LCM of two numbers is the smallest positive number that is divisible by both numbers. So, in this case, the LCM of the two numbers is 60.
Let's assume the smaller number is x. According to the given information, the larger number is 7 more than the smaller number, which means it can be expressed as x + 7.
Now, we need to find the LCM of the two numbers. One way to do this is by calculating the prime factorization of each number and then finding the highest powers of the common prime factors. However, since the numbers are not provided, we can solve this problem through a systematic approach.
We can create a table and list the multiples of the larger number (x + 7) until we find a multiple that is also a multiple of the smaller number (x).
Multiples of (x + 7):
(x + 7), 2(x + 7), 3(x + 7), 4(x + 7), ...
Multiples of x:
x, 2x, 3x, 4x, ...
We continue listing multiples for both numbers until we find a common multiple. In this case, the common multiple is 60, as given in the problem.
By examining the multiples, we find that the smaller number (x) is 20 and the larger number (x + 7) is 27.
To verify, we can check if 20 and 27 are indeed factors of 60. Since 20 and 27 are both factors of 60 (20 * 3 = 60 and 27 * 2 = 60), our answer is justified.
Therefore, the numbers are 20 and 27.