Question: How many of the positive factors of the number 36,000,000 are not perfect squares.
Any answer without an explanation is spam.
Ans.
My answer is 149.Is it right or wrong??
Here goes my explanation:
36000000=6 x 6 x 10 x 10 x 10 x 10 x 10 x 10
=2 x 3 x 2 x 3 x 2 x 5 x 2 x 5 x 2 x 5 x 2 x 5 x 2 x 5 x 2 x 5
=28 x 32 x 56
Factors possible are:
1
2, 22, 23, 24, 25, 26, 27, 28
3, 32
5, 52, 53, 54, 55, 56
For each (2 to 28 and 3 to 32 ) there will be total 16 (8 x 2) combinations of factors having subfactors 2 and 3 only.
2 x 3, 22 x 3, 23 x 3, 24 x 3, 25 x 3, 26 x 3, 27 x 3, 28 x 3
2 x 32, 22 x 32, 23 x 32, 24 x 32, 25 x 32, 26 x 32, 27 x 32, 28 x 32
Similarly for each (2 to 28 and 5 to 56 ) there will be total 48(8 x 6) combinations of factors having subfactors 2 and 5 only.
2 x 5, 22 x 5, 23 x5, 24 x 5, 25 x 5, 26 x 5, 27 x 5, 28 x 5
2 x 52, 22 x 52, 23 x 52, 24 x 52, 25 x 52, 26 x 52, 27 x 52, 28 x 52
2 x 53, 22 x 53, 23 x 53, 24 x 53, 25 x 53, 26 x 53, 27 x 53, 28 x 53
2 x 54, 22 x 54, 23 x 54, 24 x 54, 25 x 54, 26 x 54, 27 x 54, 28 x 54
2 x 55, 22 x 55, 23 x 55, 24 x 55, 25 x 55, 26 x 55, 27 x 55, 28 x 55
2 x 56, 22 x 56, 23 x 56, 24 x 56, 25 x 56, 26 x 56, 27 x 56, 28 x 56
For each (3 to 32 and 5 to 56 ) there will be total 12 (6 x 2) combinations of factors having subfactors 3 and 5 only.
3 x 5, 32 x 5
3 x 52, 32 x 52
3 x 53, 32 x 53
3 x 54, 32 x 54
3 x 55, 32 x 55
3 x 56, 32 x 56
Again for each (2 to 28 and 3 to 32 and 5 to 56 ) there will be total 96 (8 x 2 x 6) combinations of factors having subfactors 2,3 and 5 only.
2 x 3 x 5, 22x 3 x 5, 23x 3 x 5, 24 x 3 x 5, 25 x 3 x 5, 26 x 3 x 5, 27 x 3 x 5, 28 x 3 x 5
2 x 32 x 5 , 22x 32 x 5,23x 32 x 5, 24 x 32 x 5, 25 x 32 x 5, 26 x 32 x 5, 27 x 32 x 5, 28 x 32 x 5
2 x 3x 52, 22x 3 x 52, 23x 3 x 52, 24 x 3 x 52, 25 x 3 x 52, 26 x 3 x 52, 27 x 3 x 52, 28 x 3 x 52
2 x 32 x 52 , 22x 32 x 52, 23x 32 x 52, 24 x 32 x 52, 25 x 32 x 52, 26 x 32 x 52, 27 x 32 x 52, 28 x 32 x 52
2 x 3x 53, 22x 3 x 53, 23x 3 x 53, 24 x 3 x 53, 25 x 3 x 53, 26 x 3 x 53, 27 x 3 x 53, 28 x 3 x 53
2 x 32 x 53 , 22x 32 x 53, 23x 32 x 53, 24 x 32 x 53, 25 x 32 x 53, 26 x 32 x 53, 27 x 32 x 53, 28 x 32 x 53
2 x 3x 54, 22x 3 x 54, 23x 3 x 54, 24 x 3 x 54, 25 x 3 x 54, 26 x 3 x 54, 27 x 3 x 54, 28 x 3 x 54
2 x 32 x 54, 22x 32 x 54, 23x 32 x 54, 24 x 32 x 54, 25 x 32 x 54, 26 x 32 x 54, 27 x 32 x 54, 28 x 32 x 54
2 x 3x 55, 22x 3 x 55, 23x 3 x 55, 24 x 3 x 55, 25 x 3 x 55, 26 x 3 x 55, 27 x 3 x 55, 28 x 3 x 55
2 x 32 x 55 , 22x 32 x 55, 23x 32 x 55, 24 x 32 x 55, 25 x 32 x 55, 26 x 32 x 55, 27 x 32 x 55, 28 x 32 x 55
2 x 3x 56, 22x 3 x 56, 23x 3 x 56, 24 x 3 x 56, 25 x 3 x 56, 26 x 3 x 56, 27 x 3 x 56, 28 x 3 x 56
2 x 32 x 56, 22x 32 x 56, 23x 32 x 56, 24 x 32 x 56, 25 x 32 x 56, 26 x 32 x 56, 27 x 32 x 56, 28 x 32 x 56
And finally 28 x 32 x 56 itself is also a factor
So counting all those factors whose subfactor's exponential power is an even no, I get total no of factors of 36000000 which are not perfect squares as 149.
All such factors are marked in bold.
How many positive factors of 36,000,000 are not perfect squares?
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Wednesday, April 1, 2009
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