How many positive factors of 36,000,000 are not perfect squares?
Question: How many of the positive factors of the number 36,000,000 are not perfect squares.
Any answer without an explanation is spam. :smile:
Any answer without an explanation is spam. :smile:
Replies
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shalini_goel14My 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
=2[sup]8[/sup] x 3[sup]2[/sup] x 5[sup]6[/sup]
Factors possible are:
1
2, 2[sup]2[/sup], 2[sup]3[/sup], 2[sup]4[/sup], 2[sup]5[/sup], 2[sup]6[/sup], 2[sup]7[/sup], 2[sup]8[/sup]
3, 3[sup]2[/sup]
5, 5[sup]2[/sup], 5[sup]3[/sup], 5[sup]4[/sup], 5[sup]5[/sup], 5[sup]6[/sup]
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, 2[sup]2[/sup] x 3, 2[sup]3[/sup] x 3, 2[sup]4[/sup] x 3, 2[sup]5[/sup] x 3, 2[sup]6[/sup] x 3, 2[sup]7[/sup] x 3, 2[sup]8[/sup] x 3
2 x 3[sup]2[/sup], 2[sup]2[/sup] x 3[sup]2[/sup], 2[sup]3[/sup] x 3[sup]2[/sup], 2[sup]4[/sup] x 3[sup]2[/sup], 2[sup]5[/sup] x 3[sup]2[/sup], 2[sup]6[/sup] x 3[sup]2[/sup], 2[sup]7[/sup] x 3[sup]2[/sup], 2[sup]8[/sup] x 3[sup]2[/sup]
Similarly for each (2 to 2[sup]8[/sup] and 5 to 5[sup]6[/sup] ) there will be total 48(8 x 6) combinations of factors having subfactors 2 and 5 only.
2 x 5, 2[sup]2[/sup] x 5, 2[sup]3[/sup] x5, 2[sup]4[/sup] x 5, 2[sup]5[/sup] x 5, 2[sup]6[/sup] x 5, 2[sup]7[/sup] x 5, 2[sup]8[/sup] x 5
2 x 5[sup]2[/sup], 2[sup]2[/sup] x 5[sup]2[/sup], 2[sup]3[/sup] x 5[sup]2[/sup], 2[sup]4[/sup] x 5[sup]2[/sup], 2[sup]5[/sup] x 5[sup]2[/sup], 2[sup]6[/sup] x 5[sup]2[/sup], 2[sup]7[/sup] x 5[sup]2[/sup], 2[sup]8[/sup] x 5[sup]2[/sup]
2 x 5[sup]3[/sup], 2[sup]2[/sup] x 5[sup]3[/sup], 2[sup]3[/sup] x 5[sup]3[/sup], 2[sup]4[/sup] x 5[sup]3[/sup], 2[sup]5[/sup] x 5[sup]3[/sup], 2[sup]6[/sup] x 5[sup]3[/sup], 2[sup]7[/sup] x 5[sup]3[/sup], 2[sup]8[/sup] x 5[sup]3[/sup]
2 x 5[sup]4[/sup], 2[sup]2[/sup] x 5[sup]4[/sup], 2[sup]3[/sup] x 5[sup]4[/sup], 2[sup]4[/sup] x 5[sup]4[/sup], 2[sup]5[/sup] x 5[sup]4[/sup], 2[sup]6[/sup] x 5[sup]4[/sup], 2[sup]7[/sup] x 5[sup]4[/sup], 2[sup]8[/sup] x 5[sup]4[/sup]
2 x 5[sup]5[/sup], 2[sup]2[/sup] x 5[sup]5[/sup], 2[sup]3[/sup] x 5[sup]5[/sup], 2[sup]4[/sup] x 5[sup]5[/sup], 2[sup]5[/sup] x 5[sup]5[/sup], 2[sup]6[/sup] x 5[sup]5[/sup], 2[sup]7[/sup] x 5[sup]5[/sup], 2[sup]8[/sup] x 5[sup]5[/sup]
2 x 5[sup]6[/sup], 2[sup]2[/sup] x 5[sup]6[/sup], 2[sup]3[/sup] x 5[sup]6[/sup], 2[sup]4[/sup] x 5[sup]6[/sup], 2[sup]5[/sup] x 5[sup]6[/sup], 2[sup]6[/sup] x 5[sup]6[/sup], 2[sup]7[/sup] x 5[sup]6[/sup], 2[sup]8[/sup] x 5[sup]6[/sup]
For each (3 to 3[sup]2[/sup] and 5 to 5[sup]6[/sup] ) there will be total 12 (6 x 2) combinations of factors having subfactors 3 and 5 only.
3 x 5, 3[sup]2[/sup] x 5
3 x 5[sup]2[/sup], 3[sup]2[/sup] x 5[sup]2[/sup]
3 x 5[sup]3[/sup], 3[sup]2[/sup] x 5[sup]3[/sup]
3 x 5[sup]4[/sup], 3[sup]2[/sup] x 5[sup]4[/sup]
3 x 5[sup]5[/sup], 3[sup]2[/sup] x 5[sup]5[/sup]
3 x 5[sup]6[/sup], 3[sup]2[/sup] x 5[sup]6[/sup]
Again for each (2 to 2[sup]8[/sup] and 3 to 3[sup]2[/sup] and 5 to 5[sup]6[/sup] ) 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, 2[sup]2[/sup]x 3 x 5, 2[sup]3[/sup]x 3 x 5, 2[sup]4[/sup] x 3 x 5, 2[sup]5[/sup] x 3 x 5, 2[sup]6[/sup] x 3 x 5, 2[sup]7[/sup] x 3 x 5, 2[sup]8[/sup] x 3 x 5
2 x 3[sup]2[/sup] x 5 , 2[sup]2[/sup]x 3[sup]2[/sup] x 5,2[sup]3[/sup]x 3[sup]2[/sup] x 5, 2[sup]4[/sup] x 3[sup]2[/sup] x 5, 2[sup]5[/sup] x 3[sup]2[/sup] x 5, 2[sup]6[/sup] x 3[sup]2[/sup] x 5, 2[sup]7[/sup] x 3[sup]2[/sup] x 5, 2[sup]8[/sup] x 3[sup]2[/sup] x 5
2 x 3x 5[sup]2[/sup], 2[sup]2[/sup]x 3 x 5[sup]2[/sup], 2[sup]3[/sup]x 3 x 5[sup]2[/sup], 2[sup]4[/sup] x 3 x 5[sup]2[/sup], 2[sup]5[/sup] x 3 x 5[sup]2[/sup], 2[sup]6[/sup] x 3 x 5[sup]2[/sup], 2[sup]7[/sup] x 3 x 5[sup]2[/sup], 2[sup]8[/sup] x 3 x 5[sup]2[/sup]
2 x 3[sup]2[/sup] x 5[sup]2[/sup] , 2[sup]2[/sup]x 3[sup]2[/sup] x 5[sup]2[/sup], 2[sup]3[/sup]x 3[sup]2[/sup] x 5[sup]2[/sup], 2[sup]4[/sup] x 3[sup]2[/sup] x 5[sup]2[/sup], 2[sup]5[/sup] x 3[sup]2[/sup] x 5[sup]2[/sup], 2[sup]6[/sup] x 3[sup]2[/sup] x 5[sup]2[/sup], 2[sup]7[/sup] x 3[sup]2[/sup] x 5[sup]2[/sup], 2[sup]8[/sup] x 3[sup]2[/sup] x 5[sup]2[/sup]
2 x 3x 5[sup]3[/sup], 2[sup]2[/sup]x 3 x 5[sup]3[/sup], 2[sup]3[/sup]x 3 x 5[sup]3[/sup], 2[sup]4[/sup] x 3 x 5[sup]3[/sup], 2[sup]5[/sup] x 3 x 5[sup]3[/sup], 2[sup]6[/sup] x 3 x 5[sup]3[/sup], 2[sup]7[/sup] x 3 x 5[sup]3[/sup], 2[sup]8[/sup] x 3 x 5[sup]3[/sup]
2 x 3[sup]2[/sup] x 5[sup]3[/sup] , 2[sup]2[/sup]x 3[sup]2[/sup] x 5[sup]3[/sup], 2[sup]3[/sup]x 3[sup]2[/sup] x 5[sup]3[/sup], 2[sup]4[/sup] x 3[sup]2[/sup] x 5[sup]3[/sup], 2[sup]5[/sup] x 3[sup]2[/sup] x 5[sup]3[/sup], 2[sup]6[/sup] x 3[sup]2[/sup] x 5[sup]3[/sup], 2[sup]7[/sup] x 3[sup]2[/sup] x 5[sup]3[/sup], 2[sup]8[/sup] x 3[sup]2[/sup] x 5[sup]3[/sup]
2 x 3x 5[sup]4[/sup], 2[sup]2[/sup]x 3 x 5[sup]4[/sup], 2[sup]3[/sup]x 3 x 5[sup]4[/sup], 2[sup]4[/sup] x 3 x 5[sup]4[/sup], 2[sup]5[/sup] x 3 x 5[sup]4[/sup], 2[sup]6[/sup] x 3 x 5[sup]4[/sup], 2[sup]7[/sup] x 3 x 5[sup]4[/sup], 2[sup]8[/sup] x 3 x 5[sup]4[/sup]
2 x 3[sup]2[/sup] x 5[sup]4[/sup], 2[sup]2[/sup]x 3[sup]2[/sup] x 5[sup]4[/sup], 2[sup]3[/sup]x 3[sup]2[/sup] x 5[sup]4[/sup], 2[sup]4[/sup] x 3[sup]2[/sup] x 5[sup]4[/sup], 2[sup]5[/sup] x 3[sup]2[/sup] x 5[sup]4[/sup], 2[sup]6[/sup] x 3[sup]2[/sup] x 5[sup]4[/sup], 2[sup]7[/sup] x 3[sup]2[/sup] x 5[sup]4[/sup], 2[sup]8[/sup] x 3[sup]2[/sup] x 5[sup]4[/sup]
2 x 3x 5[sup]5[/sup], 2[sup]2[/sup]x 3 x 5[sup]5[/sup], 2[sup]3[/sup]x 3 x 5[sup]5[/sup], 2[sup]4[/sup] x 3 x 5[sup]5[/sup], 2[sup]5[/sup] x 3 x 5[sup]5[/sup], 2[sup]6[/sup] x 3 x 5[sup]5[/sup], 2[sup]7[/sup] x 3 x 5[sup]5[/sup], 2[sup]8[/sup] x 3 x 5[sup]5[/sup]
2 x 3[sup]2[/sup] x 5[sup]5[/sup] , 2[sup]2[/sup]x 3[sup]2[/sup] x 5[sup]5[/sup], 2[sup]3[/sup]x 3[sup]2[/sup] x 5[sup]5[/sup], 2[sup]4[/sup] x 3[sup]2[/sup] x 5[sup]5[/sup], 2[sup]5[/sup] x 3[sup]2[/sup] x 5[sup]5[/sup], 2[sup]6[/sup] x 3[sup]2[/sup] x 5[sup]5[/sup], 2[sup]7[/sup] x 3[sup]2[/sup] x 5[sup]5[/sup], 2[sup]8[/sup] x 3[sup]2[/sup] x 5[sup]5[/sup]
2 x 3x 5[sup]6[/sup], 2[sup]2[/sup]x 3 x 5[sup]6[/sup], 2[sup]3[/sup]x 3 x 5[sup]6[/sup], 2[sup]4[/sup] x 3 x 5[sup]6[/sup], 2[sup]5[/sup] x 3 x 5[sup]6[/sup], 2[sup]6[/sup] x 3 x 5[sup]6[/sup], 2[sup]7[/sup] x 3 x 5[sup]6[/sup], 2[sup]8[/sup] x 3 x 5[sup]6[/sup]
2 x 3[sup]2[/sup] x 5[sup]6[/sup], 2[sup]2[/sup]x 3[sup]2[/sup] x 5[sup]6[/sup], 2[sup]3[/sup]x 3[sup]2[/sup] x 5[sup]6[/sup], 2[sup]4[/sup] x 3[sup]2[/sup] x 5[sup]6[/sup], 2[sup]5[/sup] x 3[sup]2[/sup] x 5[sup]6[/sup], 2[sup]6[/sup] x 3[sup]2[/sup] x 5[sup]6[/sup], 2[sup]7[/sup] x 3[sup]2[/sup] x 5[sup]6[/sup], 2[sup]8[/sup] x 3[sup]2[/sup] x 5[sup]6[/sup]
And finally 2[sup]8[/sup] x 3[sup]2[/sup] x 5[sup]6[/sup] 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.
PS: Giving an explanation to others was a bigger problem rather than solving it 😀. -
becks149.
36,000,000=2^(8)*3^(2)*5^(6)
so, TOTAL NO. OF FACTORS =9*3*7=189.
NOW The factors will be PERFECT SQUARES if the EXPONENT of PRIME NUMBERS OCCURING IN THE FACTOR are EVEN.
SO TOTAL NO. OF FACTORS which are perfect squares=5*2*4=40.
As Exponent of 2 can be either of the following(0,2,4,6,8) SO 5 WAYS.SIMILARLY FOR 3 AND 5.
SO NO. OF FACTORS WHICH ARE NOT PERFECT SQUARES=189-40=149.
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