The Least Common Multiple of two numbers is 60, and one of the numbers is 7 less than the other number. What are the numbers? Justify your answer.
Factors of 60
2, 30
3, 20
4, 15
5, 12
6, 10
5 and 12
5×12=60 so they are factors of 60
12-5=7 so one number is seven less
Too cool -- don't you think that Priscilla is smart enough to figure that out herself??
Sorry, got carried away! Won't happen again, I promise.
wow!!! Ms. Sue Ain't the nice kind of person
To find the numbers, let's follow a step-by-step approach:
Step 1: Let's suppose one of the numbers is represented as x. Since the other number is 7 less than x, we can represent it as (x - 7).
Step 2: We are given that the least common multiple (LCM) of the two numbers is 60. The LCM can be found by multiplying the highest power of each prime factor that appears in the numbers. Let's determine the prime factorization of 60.
Prime factorization of 60:
60 = 2 * 2 * 3 * 5
Step 3: Now, let's check which prime factors are present in the two numbers x and (x - 7). To do this, we need to prime factorize x and (x - 7) as well.
Step 4: Let's start with x. We need to find the prime factorization of x. However, since we don't have any specific information about x, we won't be able to determine the exact prime factorization.
Step 5: Now let's consider (x - 7). We already know that the prime factorization of 60 includes 2, 2, 3, and 5. If (x - 7) has any of these prime factors, they will contribute to the LCM.
Step 6: Let's consider each prime factor separately:
Prime factor 2: If (x - 7) is divisible by 2, it means that 2 is a prime factor. Dividing (x - 7) by 2 gives (x - 7)/2. For the LCM to be valid, (x - 7)/2 must be a whole number. Therefore, (x - 7) must be divisible by 2.
Prime factor 3: Similar to the previous step, (x - 7) must be divisible by 3 for 3 to be a prime factor.
Prime factor 5: Again, (x - 7) must be divisible by 5 for 5 to be a prime factor.
Step 7: Now, let's focus on the prime factor 2. We already know that (x - 7) must be divisible by 2 to contribute to the LCM. Therefore, we can denote (x - 7) as 2y, where y is a positive integer. This way, we satisfy the condition that (x - 7)/2 is a whole number.
Step 8: Considering the prime factor 3, we know that (x - 7) must be divisible by 3. Since we represented (x - 7) as 2y in Step 7, we need to check if 2y is divisible by 3.
Step 9: If 2y is divisible by 3, it means y must be divisible by 3 as well. Let's represent y as 3z, where z is a positive integer.
Step 10: Combining the information from Steps 7 and 9, we have (x - 7) = 2y = 2(3z) = 6z.
Step 11: Finally, let's focus on the prime factor 5. We know that (x - 7) must be divisible by 5. Substituting (x - 7) with 6z, we have 6z is divisible by 5.
Step 12: To determine the possible values of z, we can divide 6z by 5 and check if it results in a whole number.
6z/5 = z(6/5)
Since we need z to be a whole number, the fraction (6/5) can be simplified to a whole number. The only way this can be accomplished is if z = 5.
Step 13: Substituting z = 5 in (x - 7) = 6z, we have:
(x - 7) = 6(5) = 30
Step 14: Adding 7 to both sides of the equation, we find:
x = 30 + 7 = 37
Step 15: Therefore, the numbers are x = 37 and (x - 7) = 30.
To summarize, the two numbers are 37 and 30, with 37 being 7 more than 30. Hence, 37 is not 7 less than the other number; it is, in fact, 7 more than the other number.