Austin must choose a number between 67 and 113 that is a multiple of 3, 8 and 12. Write all the numbers that he could choose. If there is more than one number, separate them with commas.
Multiples of 12: 72, 84, 96, 108
Now write out the multiples of 3 and 8.
Which multiples are multiples of all three numbers?
WRONG
To find the numbers between 67 and 113 that are multiples of 3, 8, and 12, we can start by finding the least common multiple (LCM) of 3, 8, and 12, which is 24.
Next, we need to find the smallest number greater than or equal to 67 that is divisible by 24. We can do this by finding the remainder when 67 is divided by 24.
67 divided by 24 equals 2 remainder 19.
So we need to add 24 - 19 to 67 to find the smallest number divisible by 24 that is greater than or equal to 67.
67 + (24 - 19) = 72.
The smallest number greater than or equal to 67 that is a multiple of 3, 8, and 12 is 72.
Now we can find the largest number that is less than or equal to 113 and divisible by 24. We can do this by finding the quotient when 113 is divided by 24 and multiplying it by 24.
113 divided by 24 equals 4 remainder 17.
So we need to multiply 24 by 4 to find the largest number divisible by 24 that is less than or equal to 113.
24 x 4 = 96.
Therefore, the numbers that Austin can choose between 67 and 113 that are multiples of 3, 8, and 12 are:
72, 96
To find the numbers that Austin could choose, we need to find the multiples of 3, 8, and 12 within the given range (67-113).
Step 1: Find the multiples of 3 within the range.
To find the multiples of 3, we need to determine the first and last multiples within the range.
The first multiple of 3 in the range:
67 is not divisible by 3, so we move to the next number, 68. 68 is not divisible by 3 either, so we continue until we find the first multiple, which is 69.
The last multiple of 3 in the range:
113 is not divisible by 3, so we move to the previous number, 112. 112 is divisible by 3, so it is the last multiple within the range.
Thus, the multiples of 3 within the range are: 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111.
Step 2: Find the multiples of 8 within the range.
To find the multiples of 8, we need to determine the first and last multiples within the range.
The first multiple of 8 in the range:
68 is not divisible by 8, so we move to the next number, 69. 69 is not divisible by 8 either, so we continue until we find the first multiple, which is 72.
The last multiple of 8 in the range:
112 is divisible by 8, so it is the last multiple within the range.
Thus, the multiples of 8 within the range are: 72, 80, 88, 96, 104, 112.
Step 3: Find the multiples of 12 within the range.
To find the multiples of 12, we need to determine the first and last multiples within the range.
The first multiple of 12 in the range:
72 is divisible by 12, so it is the first multiple within the range.
The last multiple of 12 in the range:
108 is divisible by 12, so it is the last multiple within the range.
Thus, the multiples of 12 within the range are: 72, 84, 96, 108.
Step 4: Find the common multiples.
To find the numbers that are multiples of 3, 8, and 12, we need to find the common multiples among the three sets of multiples determined in the previous steps.
The common multiples among the sets are: 72, 96.
Therefore, the numbers that Austin could choose are 72 and 96.