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.

Any help would be greatly appreciated. Thanks in advance.

http://www.jiskha.com/display.cgi?id=1409874255

To find the two numbers, let's make an assumption. Let's assume one number is x.

According to the given information, the other number is 7 less than x, so it can be expressed as (x - 7).

The least common multiple (LCM) of two numbers can be found by multiplying the two numbers and dividing it by their greatest common divisor (GCD).

So, the LCM of x and (x - 7) is given as:

LCM(x, x - 7) = (x * (x - 7)) / GCD(x, x - 7)

Since the LCM is given as 60, we can set up the equation:

(x * (x - 7)) / GCD(x, x - 7) = 60

Now, let's find the GCD of x and (x - 7) to simplify the equation further.

The GCD of two numbers does not change if we subtract one number from the other repeatedly. So, let's subtract (x - 7) from x:

GCD(x, x - 7) = GCD(7, x - 7)

By subtracting, we get:

GCD(7, x - 7) = GCD(7, 0)

Since any number divided by 0 is undefined, we conclude that:

GCD(7, x - 7) = 7

Now we can rewrite the equation:

(x * (x - 7)) / 7 = 60

Multiply both sides of the equation by 7 to eliminate the denominator:

x * (x - 7) = 420

Expand the left side of the equation:

x^2 - 7x = 420

Rearrange the equation to the standard form:

x^2 - 7x - 420 = 0

Now we need to solve this quadratic equation. Since there are no common factors, we can use factoring, completing the square, or the quadratic formula.

To find the two numbers, we need to solve the quadratic equation x^2 - 7x - 420 = 0.

Factoring the quadratic equation, we can rewrite it as:

(x - 28)(x + 15) = 0

This gives us two possibilities:

x - 28 = 0 or x + 15 = 0

Solving each equation separately, we find:

x = 28 or x = -15

Since the problem asks for positive numbers, we discard the solution x = -15.

Therefore, the value of x is 28.

Now, we can find the two numbers:

One number = x = 28
The other number = x - 7 = 28 - 7 = 21

So, the two numbers are 21 and 28.

To justify this answer, we can calculate their LCM:

LCM(21, 28) = (21 * 28) / GCD(21, 28) = (588) / 7 = 84

Since LCM(21, 28) = 84, which matches the given LCM of 60, we can conclude that the numbers are indeed 21 and 28.

To find the numbers, let's first set up equations based on the given information.

Let's assume that the two numbers are x and y, where x is greater than y.

We know that the least common multiple (LCM) of x and y is 60. LCM is the smallest multiple that both numbers share.

So, we can write the following equation:

LCM(x, y) = 60

Next, we're given that one of the numbers is 7 less than the other number. We can express this as:

x = y + 7

Now, let's simplify these equations to solve for the values of x and y.

To find the LCM of the two numbers, we need to find their factors and identify their common factors. By factoring 60, we get:

60 = 2^2 * 3 * 5

Now, we know that LCM(x, y) = 60. Since one of the numbers is 7 less than the other, we can rewrite this as:

LCM(y + 7, y) = 60

To find the LCM, we need to consider the prime factors of 60.

Multiplying the common prime factors by the highest power, we get:

LCM(y + 7, y) = 2^2 * 3 * 5

Now, equating the corresponding powers of prime factors, we have:

y + 7 = 2^2 * 3 * 5

Expanding this equation, we get:

y + 7 = 4 * 3 * 5

Simplifying further, we have:

y + 7 = 60

Subtracting 7 from both sides, we find:

y = 60 - 7
y = 53

Now that we know the value of y, we can substitute it back into the equation x = y + 7:

x = 53 + 7
x = 60

Therefore, the two numbers are 53 and 60.

To justify this answer, we can check if the LCM of 53 and 60 is indeed 60. The prime factors of 60 are 2^2, 3, and 5. By checking the common factors with powers, we find that the LCM of 53 and 60 is indeed 60, confirming our solution.