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# How many integers between 1 and 16, inclusive, have exactly 3 differen

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How many integers between 1 and 16, inclusive, have exactly 3 differen  [#permalink]

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23 Jul 2017, 09:38
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Difficulty:

55% (hard)

Question Stats:

61% (00:57) correct 39% (01:08) wrong based on 940 sessions

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How many integers between 1 and 16, inclusive, have exactly 3 different positive integer factors? (Note: 6 is NOT such an integer because 6 has 4 different positive integer factors: 1, 2, 3, and 6.)

A. 1

B. 2

C. 3

D. 4

E. 5

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Re: How many integers between 1 and 16, inclusive, have exactly 3 differen  [#permalink]

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23 Jul 2017, 22:41
5
4
carcass wrote:
How many integers between 1 and 16, inclusive, have exactly 3 different positive integer factors? (Note: 6 is NOT such an integer because 6 has 4 different positive integer factors: 1, 2, 3, and 6.)

A. 1

B. 2

C. 3

D. 4

E. 5

1. Positive integer to have odd number of factors must be a perfect square (the square of an integer).
2. Positive integer to have 3 factors must be of a form prime^2. For example, 2^2, 3^2, 5^2, ...

According to the above, we are looking for squares of primes from 1 to 16, inclusive. There are only two: 4 = 2^2 and 9 = 3^2.

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Re: How many integers between 1 and 16, inclusive, have exactly 3 differen  [#permalink]

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23 Jul 2017, 09:58
2
B
It should be a perfect square in prime factors form to have 3 diff factors
4,9 are perfect squares that hv 3 factors (excluded 1 and 16 as they have 1 and 5 factors respectively)

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Re: How many integers between 1 and 16, inclusive, have exactly 3 differen  [#permalink]

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23 Jul 2017, 09:59
9 - 1, 3, 9
4 - 1, 2, 4

Ans B

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How many integers between 1 and 16, inclusive, have exactly 3 differen  [#permalink]

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23 Jul 2017, 22:43
3
3
Bunuel wrote:
carcass wrote:
How many integers between 1 and 16, inclusive, have exactly 3 different positive integer factors? (Note: 6 is NOT such an integer because 6 has 4 different positive integer factors: 1, 2, 3, and 6.)

A. 1

B. 2

C. 3

D. 4

E. 5

1. Positive integer to have odd number of factors must be a perfect square (the square of an integer).
2. Positive integer to have 3 factors must be of a form prime^2. For example, 2^2, 3^2, 5^2, ...

According to the above, we are looking for squares of primes from 1 to 16, inclusive. There are only two: 4 = 2^2 and 9 = 3^2.

To elaboratre more:

Finding the Number of Factors of an Integer:

First make prime factorization of an integer $$n=a^p*b^q*c^r$$, where $$a$$, $$b$$, and $$c$$ are prime factors of $$n$$ and $$p$$, $$q$$, and $$r$$ are their powers.

The number of factors of $$n$$ will be expressed by the formula $$(p+1)(q+1)(r+1)$$. NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: $$450=2^1*3^2*5^2$$

Total number of factors of 450 including 1 and 450 itself is $$(1+1)*(2+1)*(2+1)=2*3*3=18$$ factors.

1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;

2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. The reverse is also true: if a number has an ODD number of Odd-factors, and EVEN number of Even-factors then it's a perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor) and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors);

4. Perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: $$36=2^2*3^2$$, powers of prime factors 2 and 3 are even.

Hope it helps.
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Re: How many integers between 1 and 16, inclusive, have exactly 3 differen  [#permalink]

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05 Sep 2017, 05:56
carcass wrote:
How many integers between 1 and 16, inclusive, have exactly 3 different positive integer factors? (Note: 6 is NOT such an integer because 6 has 4 different positive integer factors: 1, 2, 3, and 6.)

A. 1

B. 2

C. 3

D. 4

E. 5

Perfect Squares always have Odd number of factors.
Three is an odd number the question specifies.
Excluding 1 and 16, there are only two perfect squares between 1 - 16 -- 4,9

4 = (4,2,1) = three factors
9 = (9,3,1) = three factors

Therefore, there is two integers with 3 different positive integer factors.

Ans (B) 2
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Re: How many integers between 1 and 16, inclusive, have exactly 3 differen  [#permalink]

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16 Oct 2017, 11:02
HUM... I missed the question.
True, 16 has mor than 3 factors. it has 5 factors
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Re: How many integers between 1 and 16, inclusive, have exactly 3 differen  [#permalink]

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08 Aug 2018, 21:08
1
Could also use a more blunt force approach and still get to the answer is well under 2 minutes (in case someone doesn't intuitively make the perfect square and odd # of factors connection right away)

List #s ---> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16
Eliminate all primes (b/c only two unique factors i.e. the definition of a prime) ---> 1, 4, 6, 8, 9, 10, 12, 14, 15, 16
Quickly run through the factors in your head (or write out on scratch pad if desired) ---> left with only 4 and 9 i.e 2 #s

PS. the tags on this post have the question labelled as a 700-level but seems like it would be more accurately categorized as sub-600? As a point of reference, the question appears in the 2018 quant review book as #52 (pretty early hence lower level)
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Re: How many integers between 1 and 16, inclusive, have exactly 3 differen  [#permalink]

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19 Aug 2018, 20:27
First of all skip all prime numbers as they have exactly 2 positive integer factors. So we are left with 4,6,8,9,10,12,14,15,16
By finding factors of all these we can conclude that only 2 numbers (4 and 9) have exactly 3 positive integer factors.

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Re: How many integers between 1 and 16, inclusive, have exactly 3 differen  [#permalink]

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23 Aug 2018, 15:41
carcass wrote:
How many integers between 1 and 16, inclusive, have exactly 3 different positive integer factors? (Note: 6 is NOT such an integer because 6 has 4 different positive integer factors: 1, 2, 3, and 6.)

A. 1

B. 2

C. 3

D. 4

E. 5

An integer of the form p^2 where p is a prime will have exactly 3 factors. Thus 2^2 = 4 and 3^2 = 9 are the only two integers between 1 and 16, inclusive, that have exactly 3 factors.

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Re: How many integers between 1 and 16, inclusive, have exactly 3 differen  [#permalink]

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19 Oct 2018, 04:28
carcass wrote:
How many integers between 1 and 16, inclusive, have exactly 3 different positive integer factors? (Note: 6 is NOT such an integer because 6 has 4 different positive integer factors: 1, 2, 3, and 6.)

A. 1

B. 2

C. 3

D. 4

E. 5

This is basically asking for a prime perfect square. Because only a prime perfect square has 3 different positive factors

2^2 = 4 (4,2,1)
3^2 = 9 (9,3,1)
5^2 = 25 (25,5,1)

However, here we can only have 2 and 3. so only 2 integers.

Re: How many integers between 1 and 16, inclusive, have exactly 3 differen &nbs [#permalink] 19 Oct 2018, 04:28
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