Kaitlin must choose a number between
49
and
95
that is a multiple of
3
,
6
, and
7
. Write all the numbers that she could choose. If there is more than one number, separate them with commas.
LCM(3,6,7) = 42
so, we want all multiples of 42 between 49 and 95
That means only 84
Well, well, Kaitlin is a lucky one! Let me tell you, she has quite a few options to choose from. Here are the numbers that she could select: 51, 57, 63, 69, 75, 81, 87, and 93. So many choices, I hope Kaitlin can handle the pressure!
To find the numbers between 49 and 95 that are multiples of 3, 6, and 7, we need to find the numbers that are divisible by all three of these numbers.
First, let's find the numbers that are multiples of 3 between 49 and 95.
49 divided by 3 equals 16 with a remainder of 1.
50 divided by 3 equals 16 with a remainder of 2.
51 divided by 3 equals 17 with a remainder of 0.
...
95 divided by 3 equals 31 with a remainder of 2.
Since the remainder is 0 for 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87 and 90, these numbers are multiples of 3.
Now, let's check which of these numbers are also multiples of 6.
51 divided by 6 equals 8 with a remainder of 3.
...
90 divided by 6 equals 15 with a remainder of 0.
Since the remainder is 0 for 54, 60, 66, 72, 78, 84, and 90, these numbers are multiples of both 3 and 6.
Finally, let's check which of these numbers are also multiples of 7.
54 divided by 7 equals 7 with a remainder of 5.
...
90 divided by 7 equals 12 with a remainder of 6.
Since the remainder is 0 for 63 and 84, these numbers are multiples of 3, 6, and 7.
Therefore, the numbers Kaitlin could choose are 63 and 84.
To find the numbers between 49 and 95 that are multiples of 3, 6, and 7, we can determine the least common multiple (LCM) of these three numbers, which is the smallest number that is divisible by all three. In this case, the LCM of 3, 6, and 7 is 42.
Now, let's identify all the multiples of 42 between 49 and 95. To do this, divide 49 by 42, which gives us 1 with a remainder of 7. Since the remainder is greater than 0, we need to find the next multiple of 42. We then multiply 42 by 2, which gives us 84. As 84 is within the given range (49 to 95), we include it as a possible number.
Next, we divide 84 by 42 again, yielding 2 with no remainder. Since 84 is already included, we move to the next multiple of 42, which is 126. Since 126 is greater than 95, we stop here and have completed the calculation.
Therefore, the numbers Kaitlin could choose are 84.