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

Question

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

Bunuel · Accepted Answer

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.

Answer: B.

Bunuel · Answer

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.

Tips about the perfect square :

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.

prachi18oct · Answer

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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Ejiroosa · Answer

9 - 1, 3, 9
4 - 1, 2, 4

Ans B

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CyberStein · Answer

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

guialain · Answer

HUM... I missed the question.
True, 16 has mor than 3 factors. it has 5 factors

MonkeyWork21 · Answer

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

Answer: B

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)

hasnain3047 · Answer

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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ScottTargetTestPrep · Answer

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.

Answer: B

Salsanousi · Answer

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.

answer choice B

Basshead · Answer

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.)

We can eliminate prime numbers right away since they have 2 factors.

We are left with: 1, 4, 6, 8, 9, 10, 12, 14, 15, 16

We can quickly eliminate 1 and all the numbers 10 or more.

We are left with: 4, 6, 8, 9

6 and 8 can be eliminated because they have 4 factors.

We see that only 4 and 9 have exactly 3 positive factors. Answer is B.

Snehaaaaa · Answer

A perfect square of prime number has 3 different positive factors.
eg. 2^2, 3^2, 5^2
Here, in the range from 1 to 16 we have only 2 such numbers- 2^2=4; 3^2=9
Correct answer is B

lavanya.18 · Answer

We need to remember here that only perfect squares have 3 integer factors. So between 1 to 16 we only have 2 perfect squares ie. 4 and 9. Hence, our answer is 2 here.

bumpbot · Answer

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

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

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