SOLUTION: You were assigned a locker for your books at school. You forgot the locker number but remmeber that two of the 12 positive factors of the locker number are 6 and 25. What is your l

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Question 565018: You were assigned a locker for your books at school. You forgot the locker number but remmeber that two of the 12 positive factors of the locker number are 6 and 25. What is your locker number? Show or explain how you got your answer.
Answer by KMST(5398) About Me  (Show Source):
You can put this solution on YOUR website!
If two of the positive factors of the number are
6=2%2A3 and 25=5%5E2 ,
the number must be a multiple of 2%2A3%2A5%5E2=150
The number 150 has 2%2A2%2A3=12 positive factors.
Any other number who has 6 and 25 as factors will be a multiple of 150, and will have more than 12 factors.
So your locker is number 150.
How do I know 2%2A3%2A5%5E2=150 has 2%2A2%2A3=12 positive factors?
Because all the factors will be of the form
2%5Ea%2A3%5Eb%2A5%5Ec where a and b could be 0 or 1, and c could be 0, 1, or 2.
That gives you 2 choices for a, 2 choices for b,and 3 choices for c. That makes 2%2A2%2A3=12 combinations.
In case you don't believe, I'll list the factors of 150
1=1%2A1%2A1=2%5E0%2A3%5E0%2A5%5E0 2=2%2A1%2A1=2%5E1%2A3%5E0%2A5%5E0
3=1%2A3%2A1=2%5E0%2A3%5E1%2A5%5E0 6=2%2A3%2A1=2%5E1%2A3%5E1%2A5%5E0
5=1%2A1%2A5=2%5E0%2A3%5E0%2A5%5E1 10=2%2A1%2A5=2%5E1%2A3%5E0%2A5%5E1
15=1%2A3%2A5=2%5E0%2A3%5E1%2A5%5E1 30=2%2A3%2A5=2%5E1%2A3%5E1%2A5%5E1
25=1%2A1%2A25=2%5E0%2A3%5E0%2A5%5E2 50=2%2A1%2A25=2%5E1%2A3%5E0%2A5%5E2
75=1%2A3%2A25=2%5E0%2A3%5E1%2A5%5E2 150=2%2A3%2A25=2%5E1%2A3%5E1%2A5%5E2