It is currently 13 Dec 2017, 05:16

# Decision(s) Day!:

CHAT Rooms | Ross R1 | Kellogg R1 | Darden R1 | Tepper R1

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

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

Author Message
TAGS:

### Hide Tags

Board of Directors
Joined: 01 Sep 2010
Posts: 3421

Kudos [?]: 9486 [1], given: 1203

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

### Show Tags

23 Jul 2017, 09:38
1
KUDOS
Top Contributor
9
This post was
BOOKMARKED
00:00

Difficulty:

55% (hard)

Question Stats:

53% (01:02) correct 47% (00:53) wrong based on 352 sessions

### HideShow timer Statistics

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
[Reveal] Spoiler: OA

_________________

Kudos [?]: 9486 [1], given: 1203

Manager
Joined: 20 Jun 2014
Posts: 53

Kudos [?]: 14 [1], given: 24

GMAT 1: 630 Q49 V27
GMAT 2: 660 Q49 V32
Re: How many integers between 1 and 16, inclusive, have exactly 3 differen [#permalink]

### Show Tags

23 Jul 2017, 09:58
1
KUDOS
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)

Sent from my Redmi Note 4 using GMAT Club Forum mobile app

Kudos [?]: 14 [1], given: 24

Intern
Joined: 27 May 2017
Posts: 13

Kudos [?]: 2 [0], given: 0

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

### Show Tags

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

Ans B

Sent from my SM-G935F using GMAT Club Forum mobile app

Kudos [?]: 2 [0], given: 0

Math Expert
Joined: 02 Sep 2009
Posts: 42577

Kudos [?]: 135466 [0], given: 12695

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

### Show Tags

23 Jul 2017, 22:41
Expert's post
1
This post was
BOOKMARKED
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.

_________________

Kudos [?]: 135466 [0], given: 12695

Math Expert
Joined: 02 Sep 2009
Posts: 42577

Kudos [?]: 135466 [0], given: 12695

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

### Show Tags

23 Jul 2017, 22:43
Expert's post
1
This post was
BOOKMARKED
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.
_________________

Kudos [?]: 135466 [0], given: 12695

Manager
Joined: 21 Jun 2017
Posts: 71

Kudos [?]: 5 [0], given: 2

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

### Show Tags

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

Kudos [?]: 5 [0], given: 2

Manager
Joined: 01 Dec 2016
Posts: 122

Kudos [?]: 14 [0], given: 32

Location: Cote d'Ivoire
Concentration: Finance, Entrepreneurship
GMAT 1: 650 Q47 V34
WE: Investment Banking (Investment Banking)
Re: How many integers between 1 and 16, inclusive, have exactly 3 differen [#permalink]

### Show Tags

16 Oct 2017, 11:02
HUM... I missed the question.
True, 16 has mor than 3 factors. it has 5 factors
_________________

What was previously considered impossible is now obvious reality.
In the past, people used to open doors with their hands. Today, doors open "by magic" when people approach them

Kudos [?]: 14 [0], given: 32

Re: How many integers between 1 and 16, inclusive, have exactly 3 differen   [#permalink] 16 Oct 2017, 11:02
Display posts from previous: Sort by