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57%
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Is square root of a positive integer x an integer?
(1) The sum of distinct factors of x is odd
(2) x has an odd number of distinct factor
(1) The sum of distinct factors of x is odd
(2) x has an odd number of distinct factor
03 Jun 2013, 07:29
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summer101
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.
Similar questions to practice:
is-the-positive-integer-n-a-perfect-square-94700.html
is-the-positive-integer-n-a-perfect-square-1-the-number-79108.html
which-of-the-following-must-be-true-if-the-square-root-of-x-144504.html
if-the-integer-n-has-exactly-three-positive-divisors-includ-145985.html
if-x-is-an-integer-that-has-exactly-three-positive-divisors-132710.html
Hope it helps.
General Discussion
05 Jun 2013, 03:07
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Is square root of a positive integer x an integer?
Is \(\sqrt{x}=integer\)? --> is \(x=integer^2\)? So, the question basically asks whether x is a perfect square.
(1) The sum of distinct factors of x is odd. If x=1 (1 has one distinct factor: 1), then x IS a prefect square but if x=2 (2 has two distinct factor: 1 and 2 --> 1+2=3=odd), then x is NOT a prefect square. Not sufficient.
(2) x has an odd number of distinct factor. According to the properties above, x must be a perfect square. Sufficient.
Answer: B.
Hope it's clear.
Is \(\sqrt{x}=integer\)? --> is \(x=integer^2\)? So, the question basically asks whether x is a perfect square.
(1) The sum of distinct factors of x is odd. If x=1 (1 has one distinct factor: 1), then x IS a prefect square but if x=2 (2 has two distinct factor: 1 and 2 --> 1+2=3=odd), then x is NOT a prefect square. Not sufficient.
(2) x has an odd number of distinct factor. According to the properties above, x must be a perfect square. Sufficient.
Answer: B.
Hope it's clear.
