Author 
Message 
TAGS:

Hide Tags

Board of Directors
Joined: 01 Sep 2010
Posts: 3420

How many integers between 1 and 16, inclusive, have exactly 3 differen [#permalink]
Show Tags
23 Jul 2017, 10:38
Question Stats:
54% (00:59) correct 46% (00:56) wrong based on 533 sessions
HideShow timer Statistics



Manager
Joined: 20 Jun 2014
Posts: 52
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, 10:58
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



Intern
Joined: 27 May 2017
Posts: 12

Re: How many integers between 1 and 16, inclusive, have exactly 3 differen [#permalink]
Show Tags
23 Jul 2017, 10:59
9  1, 3, 9 4  1, 2, 4 Ans B Sent from my SMG935F using GMAT Club Forum mobile app



Math Expert
Joined: 02 Sep 2009
Posts: 46284

Re: How many integers between 1 and 16, inclusive, have exactly 3 differen [#permalink]
Show Tags
23 Jul 2017, 23:41
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 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.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Math Expert
Joined: 02 Sep 2009
Posts: 46284

How many integers between 1 and 16, inclusive, have exactly 3 differen [#permalink]
Show Tags
23 Jul 2017, 23:43
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. Answer: B. 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 Oddfactors, and EVEN number of Evenfactors. The reverse is also true: if a number has an ODD number of Oddfactors, and EVEN number of Evenfactors 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.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Manager
Joined: 21 Jun 2017
Posts: 78

Re: How many integers between 1 and 16, inclusive, have exactly 3 differen [#permalink]
Show Tags
05 Sep 2017, 06:56
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



Manager
Joined: 01 Dec 2016
Posts: 118
Concentration: Finance, Entrepreneurship
WE: Investment Banking (Investment Banking)

Re: How many integers between 1 and 16, inclusive, have exactly 3 differen [#permalink]
Show Tags
16 Oct 2017, 12:02
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




Re: How many integers between 1 and 16, inclusive, have exactly 3 differen
[#permalink]
16 Oct 2017, 12:02






