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# How positive integers less than 100 have exactly 4 odd

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GMAT Instructor
Joined: 04 Jul 2006
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How positive integers less than 100 have exactly 4 odd [#permalink]  11 Jul 2006, 01:45
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How positive integers less than 100 have exactly 4 odd factors but no even factors?

(A) 13 (B) 14 (C) 15 (D) 16 (E) 17

(Please don't look at all positive integers less than 100 one by one)
Director
Joined: 13 Nov 2003
Posts: 793
Location: BULGARIA
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Kudos [?]: 27 [0], given: 0

When integers in each of these triplets are multiplied we will get numbers less than 100 with 4 ODD factors each
1,3,5 1,5,7 1,7,11
1,3,9 1,5,11 1,7,13
1,3,7 1,5,13
1,3,11 1,5,17
1,3,13 1,5,19
1,3,17
1,3,19
1,3,23
1,3,29
1,3,31
I will try to explain my logic here. If all factors are odd then the numbers themselves are odd. So we narrow down the scope to 50 numbers. The number 1 will present in each and the number itself is also a factor. So we need 2 more factors. When they are primes then we need no more. The only exception is 27. Starting from the LEAST ODD PRIME(3) think that the ans is E
GMAT Instructor
Joined: 04 Jul 2006
Posts: 1269
Followers: 23

Kudos [?]: 158 [0], given: 0

Excellent! As you point out, an integer has exactly four factors if and only if it is the product of two different prime numbers or the cube of a prime number.

Looking at the odd prime numbers {3,5,7,11,13,17,19,23,29,31...}

For a product less than 100

Case I 3 multiplied by any of {5,7,11,13...31}....... 9 numbers
Case II 5 "............................ " {7,...,19}................. 5 numbers
Case III 7 " ..............................." {11,13} ............... 2 numbers

Case IV 3*3*3=27 ............................................... 1 number

TOTAL: 17
OA: E

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Last edited by kevincan on 11 Jul 2006, 06:42, edited 1 time in total.
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