what are the factors of the number 2400? and how many are there in total?
Break 2400 down into prime factors with a factor tree:
2400 = 2·2·2·2·2·3·5·5
It's easy to answer the last question. There are only
3 prime factors of 2400. These are 2,3,and 5. So
every factor is of the form
2P3Q5R
where P can take on values 0,1,2,3,4,5, that's 6 factors
Q can take on values 0,1, that's 2 factors
R can take on values 0,1,2, that's 3 factors
So that's a total of 6·2·3 = 36 factors.
But to get all those 36 factors is a lot of busy work:
We make all the combinations of those factors:
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Combinations of one factor:
1, 2, 3, and 5
This accounts for 4 factors
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Combinations of two factors:
2·2, 2·3, 2·5, 3·5. 5·5 or
This accounts for 5 factors
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Combination of three factors:
2·2·2, 2·2·3, 2·2·5, 2·3·5, 2·5·5, 3·5·5
This accounts for 6 factors
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Combinations of four factors:
2·2·2·2, 2·2·2·3, 2·2·2·5, 2·2·3·5, 2·2·5·5, 2·3·5·5
This accounts for 6 factors
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Combinations of five factors:
2·2·2·2·2, 2·2·2·2·3, 2·2·2·2·5, 2·2·2·3·3, 2·2·2·3·5, 2·2·2·5·5, 2·2·3·5·5
This accounts for 7 factors
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Combinations of six factors:
2·2·2·2·2·3, 2·2·2·2·2·5, 2·2·2·2·3·5, 2·2·2·3·5·5
This accounts for 4 factors
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Combinations of seven factors:
2·2·2·2·2·3·5, 2·2·2·2·2·5·5, 2·2·2·2·3·5·5
This accounts for 3 factors
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Combinations of eight factors:
2·2·2·2·2·3·5·5
This accounts for 1 factor.
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That's 4 + 5 + 6 + 6 + 7 + 4 + 3 + 1 = 36 factors, so we have them
all. These are, in order:
1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 32, 40, 48, 50,
60, 75, 80, 96, 100, 120, 150, 160, 200, 240, 300, 400, 480, 600, 800,
1200, 2400
Edwin
