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If d is a positive integer, is \(\sqrt{d}\) an integer?
(1) d has an odd number of positive divisors.
(2) d has 3 positive divisors.
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(1) d has an odd number of positive divisors.
(2) d has 3 positive divisors.
If positive integer d equal to the square of an integer?
(1) d has an odd number of divisors.
(2) d has 3 divisors.
(1) d has an odd number of divisors.
(2) d has 3 divisors.
Pure algebraic questions like this one are no longer part of the GMAT Data Sufficiency syllabus.
For relevant Data Sufficiency questions, please check the Data Sufficiency forum:
Data Sufficiency Forum
Data Sufficiency Forum
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RaghavKhanna
Joined: 03 May 2020
Last visit: 23 Aug 2025
Posts: 263
Location: India
Concentration: Marketing, Strategy
GMAT 1: 700 Q47 V39
GMAT 1: 700 Q47 V39
Posts: 263
If d is a positive integer, (d)^1/2 is an integer if d all prime factors of d have even power.
St. 1:
d = a^p * b^q .....
=> Number of factors of d = (p+1)(q+1)...
If number of factors = odd
=> (p+1)(q+1)... = odd
=> p, q ... are even
=> d = a^2n * b^2m...
=> d^1/2 = a^n * b^m...
=> d^1/2 is an integer.
=> St. 1 is sufficient.
St. 2:
d has three positive divisors
=> d = a^2
=> d^1/2 is an integer
=> St. 2 is sufficient.
Answer: D
Posted from my mobile device
St. 1:
d = a^p * b^q .....
=> Number of factors of d = (p+1)(q+1)...
If number of factors = odd
=> (p+1)(q+1)... = odd
=> p, q ... are even
=> d = a^2n * b^2m...
=> d^1/2 = a^n * b^m...
=> d^1/2 is an integer.
=> St. 1 is sufficient.
St. 2:
d has three positive divisors
=> d = a^2
=> d^1/2 is an integer
=> St. 2 is sufficient.
Answer: D
Posted from my mobile device
VIkash1073
Joined: 01 Jan 2020
Last visit: 29 Aug 2026
Posts: 3
Location: India
Schools: Broad Darden '24 ISB '23 Mendoza Tepper '24
GMAT 1: 680 Q49 V32
If d is a positive integer, is an integer?
(1) d has an odd number of positive divisors.
(2) d has 3 positive divisors.
Statement 1:
As only perfect squares have odd no. of divisors, hence sufficient.
Statement 2:
3 positive divisors - again odd no. of divisors, perfect square.
Example: 4 - divisors (1,2,4)
9- divisors (1,3,9)
25 - divisors (1,5,25) and so on
Sufficient.
Hence (D)
Posted from my mobile device
(1) d has an odd number of positive divisors.
(2) d has 3 positive divisors.
Statement 1:
As only perfect squares have odd no. of divisors, hence sufficient.
Statement 2:
3 positive divisors - again odd no. of divisors, perfect square.
Example: 4 - divisors (1,2,4)
9- divisors (1,3,9)
25 - divisors (1,5,25) and so on
Sufficient.
Hence (D)
Posted from my mobile device
yashikaaggarwal
Senior Moderator - Masters Forum
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Schools: Cranfield (A) Warwick MiM "23 (M)
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A perfect square had odd number of divisors.
For ex :- 4 has 3 divisors. 2^2 = (2+1) = 3
9 has 3 divisors, 3^2 = (2+1) = 3
And so on..
So both statements are sufficient alone..
IMO D
Posted from my mobile device
_________________
For ex :- 4 has 3 divisors. 2^2 = (2+1) = 3
9 has 3 divisors, 3^2 = (2+1) = 3
And so on..
So both statements are sufficient alone..
IMO D
Posted from my mobile device
_________________
I guess it is D
for 1: if d has 1 factor it is 1. root1 is an integer. if d has 3 divisors it is in the form of X^2. so x is an integer. if it has 5 divisors it is in the form of x^4, so X^2 (root of X^4) is an integer.
for 2: if d has 3 factors it is in the form of x^2. SO x is an integer.
for 1: if d has 1 factor it is 1. root1 is an integer. if d has 3 divisors it is in the form of X^2. so x is an integer. if it has 5 divisors it is in the form of x^4, so X^2 (root of X^4) is an integer.
for 2: if d has 3 factors it is in the form of x^2. SO x is an integer.
Posts: 1,052
For Statement 1 & 2,
Only perfect squares have odd no. of divisors
Both are individually sufficient
IMO-D
Only perfect squares have odd no. of divisors
Both are individually sufficient
IMO-D
Bunuel
Question: is √d and Integer = Is d a Perfect square?
CONCEPT: Perfect Sqaures {1, 4, 9, 16, 25...} always have ODD number of factors
Statement 1: d has an odd number of positive divisors.
ie.. d is a perfect square
SUFFICIENT
Statement 2: d has 3 positive divisors.
i.e. d is square of a Prime number such as 4, 9, 25, 49... etc
SUFFICIENT
ANswer: Option D
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