the third common multiple of two number is 60.find the lowest common multiple of the numbers
The lowest common multiple of the two numbers is 60.
Ah, finding the lowest common multiple, a classic mathematical quest! Let me put on my funny hat and solve this for you.
To find the lowest common multiple, we need to find the smallest number that both numbers can evenly divide into, like a perfectly sliced pizza at a math-themed party!
So, let's call the numbers A and B. We know that their third common multiple is 60. But what are the numbers? Well, we don't know yet, but hold on to your clown shoes, because we're about to find out!
To get there, we'll play a little game of detective. Let's consider that the first common multiple of A and B is X. The second common multiple would be 2 times X, and the third common multiple is 3 times X, which we know is 60.
So, 3 times X equals 60, right? That means X is equal to 60 divided by 3, which is - let me do some quick math - 20!
Fantastic! We found X, but what we really want is the lowest common multiple. Well, what better way to find it than by multiplying A and B by X?
Therefore, the lowest common multiple of the numbers A and B is simply A times X (which is 20) multiplied by B times X (also 20).
So, the formula is: Lowest Common Multiple = A × X × B × X, which simplifies to A times B times X squared.
I hope this rather amusing explanation clears things up, my friend! Happy calculating!
To find the lowest common multiple (LCM) of two numbers, we need to consider their prime factors.
Let's assume the two numbers are A and B.
First, let's find the prime factorization of 60:
60 = 2^2 × 3 × 5
Now, since the third common multiple of A and B is 60, it means that A and B both have 60 as a multiple. Thus, the numbers can be represented as multiples of 60:
A = 60 × a
B = 60 × b
Now, we need to find the prime factorization of A and B.
If we divide both A and B by the greatest common divisor (GCD) of A and B, we will get the smallest possible values for a and b. Therefore, the LCM of A and B will be:
LCM(A, B) = (GCD(A, B)) × (a × b)
Since we know that the third common multiple of A and B is 60, let's divide both A and B by the GCD to find the values of a and b.
A = 60 × a
B = 60 × b
So if the third common multiple is 60, it means that a × b must equal 1.
Therefore, the lowest common multiple (LCM) of A and B is the same as 60.
LCM(A, B) = 60
To find the lowest common multiple (LCM) of two numbers, we can use the prime factorization method.
Let's assume the two numbers are x and y.
1. Find the prime factorization of both numbers.
For example, let's say x = 2^a * 3^b * 5^c and y = 2^d * 3^e * 5^f.
Here, a, b, c, d, e, and f represent the powers of each respective prime number.
2. Identify the highest power of each prime number.
Scan through the prime factorizations of both numbers and identify the highest power for each prime number. Take the maximum value for each power in both factorizations.
3. Multiply the prime factors.
Multiply all the prime factors together, raised to their respective powers.
In our example, the LCM would be 2^max(a, d) * 3^max(b, e) * 5^max(c, f).
Now, let's find the LCM using the information provided.
If the third common multiple of the two numbers is 60, it means that both numbers are divisible by 60.
So, the prime factorization of 60 is 2^2 * 3^1 * 5^1.
Using the information provided, we can set up equations based on our prime factorizations.
x = 2^a * 3^b * 5^c
y = 2^d * 3^e * 5^f
We know that x is divisible by 60, so x = 2^2 * 3^1 * 5^1.
Now we can solve for y using the information given.
To find the LCM, we need to find the highest power for each prime number.
From the information given, we know that 2^2 is the highest power of 2, 3^1 is the highest power of 3, and 5^1 is the highest power of 5.
Therefore, the LCM of x and y would be 2^2 * 3^1 * 5^1, which simplifies to 60.
So, the lowest common multiple of the two numbers is 60.