Find six consecutive multiples of 6 whose sum is the least common multiple of 13 and 18.
To find six consecutive multiples of 6, we can start by finding the least common multiple (LCM) of 13 and 18.
Step 1: Find the prime factorization of both 13 and 18.
- Prime factorization of 13: 13
- Prime factorization of 18: 2 * 3^2
Step 2: Take the highest power of each prime factor present in either number.
- Prime factorization of LCM(13, 18): 2 * 3^2 * 13
Step 3: Multiply the prime factors together to find the LCM.
- LCM(13, 18) = 2 * 3^2 * 13 = 468
Now that we have the LCM of 13 and 18, which is 468, we need to find six consecutive multiples of 6 whose sum equals 468.
Let's say the first number in the sequence is x.
Then the sequence would be: x, x+6, x+12, x+18, x+24, x+30
The sum of these six numbers is:
x + (x+6) + (x+12) + (x+18) + (x+24) + (x+30)
Now we can set up an equation:
x + (x+6) + (x+12) + (x+18) + (x+24) + (x+30) = 468
Simplifying the equation:
6x + 90 = 468
Subtracting 90 from both sides:
6x = 468 - 90
6x = 378
Dividing both sides by 6:
x = 378 / 6
x = 63
Now we know that the first number in the sequence is 63. The six consecutive multiples of 6 are:
63, 69, 75, 81, 87, 93
And their sum is indeed equal to the LCM of 13 and 18, which is 468.
let the 6 consecutive multiples of 6 are
6x , 6x+6 , 6x+12, 6x+18, 6x+24 and 6x+30
their sum is 36x + 90
the LCM of 13 and 18 is 234
36x + 90 = 234
36x = 144
x = 4
So the number are :
24, 30, 36, 42, 48, and 54