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

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

Ans B

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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

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)

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

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

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.
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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

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