Derek R. answered • 08/28/16
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When divided by 13, there is no remainder. Thus, the number is a multiple of thirteen.
Look at the numbers 3, 5, 6, 8, 10, and 12 and determine the LCM. I do it by prime factorization.
3 - 3
5 - 5
6 - 2*3
8 - 2*2*2
10 - 2*5
12 - 2*2*3
Observe the number 2*2*2*3*5. I can take factors of that number to create all the numbers above, 3,5,6,8,10, and 12.
2*2*2*3*5 = 120
Now each of those numbers leave a remainder of 2. So, 120k+2 is a factor of 13 (where k is an integer).
Plug in the following integers into 120k+2
k = 1, 2, 3, 4, 5, ...
122, 242, 362, 482, 602, 722, 842, 962
962 is the first number in the list that is divisible by 13. 962 = 120*8 + 2. Therefore, 962 is the correct answer.
Derek R.
At the step 120k+2, list your multiples of 13. The multiple of 13 closest to 120 is 117.
120 k + 2 = 117k + (3k + 2).
If 120k + 2 is divisible by 13 and so is 117k, then 3k + 2 must be.
Plug in integers for k in 3k + 2 starting with 1. The list is as follows:
5, 8, 11, 14, 17, 20, 23, 26
26 is divisible by 13 and is obtained by plugging in 8. (26 = 3*8 + 2)
So 8 is the magic number for k. 120 * 8 + 2 = 962. I wish I knew an easier way. This way involves smaller numbers, but still heavy on calculation.
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08/28/16
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08/28/16