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Which of the following statements is/are true?
a. A perfect square always has an odd numbers of factors.
b. If a number n has five factors between 1 and the square root of n (n is a perfect square), then n has 13 factors.
c. 72 has 11 factors.
b. If a number n has five factors between 1 and the square root of n (n is a perfect square), then n has 13 factors.
c. 72 has 11 factors.
- a)(a) and (b)
- b)(b) and (c)
- c)(a) and (c)
- d)All the statements are true
- e)None of the statements is true
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Which of the following statements is/are true?a. A perfect square alwa...
A perfect square is of the form n2a
∴ Number of factors = (2a + 1), which is odd
Hence, statement (a) is correct.
Hence, options 2 and 5 can be eliminated.
Since statement (c) is easier to check using a direct formula, check for statement (c) first.
72 = 23 x 32
Number of factors = (3 + 1)(2 + 1) = 12
Thus, statement (c) is incorrect.
Hence, options 3 and 4 can be eliminated.
Hence, option 1.
Note: Statement (a) can also be checked by considering 2 or 3 random perfect squares and finding their number of factors.
Statement (b) can be verified as under:
N is a perfect square and has 5 factors between 1 and √n.
Half of the factors of a number which is a perfect square are lesser than its square root and half of them are greater than its square root.
e.g., 36 has the factors 1, 2, 3, 4, 6, 9, 12, 18 and 36 and √36 = 6
1,2, 3 and 4 are less than 6 while 9, 12, 18 and 36 are greater than 6
∴ If the number of factors of n between 1 and √n = 5, then the number of factors between √n and n is also 5.
Apart from this, 1, √n and n are also factors of n.
Total number of factors = 5 + 5 + 3=13
∴ Number of factors = (2a + 1), which is odd
Hence, statement (a) is correct.
Hence, options 2 and 5 can be eliminated.
Since statement (c) is easier to check using a direct formula, check for statement (c) first.
72 = 23 x 32
Number of factors = (3 + 1)(2 + 1) = 12
Thus, statement (c) is incorrect.
Hence, options 3 and 4 can be eliminated.
Hence, option 1.
Note: Statement (a) can also be checked by considering 2 or 3 random perfect squares and finding their number of factors.
Statement (b) can be verified as under:
N is a perfect square and has 5 factors between 1 and √n.
Half of the factors of a number which is a perfect square are lesser than its square root and half of them are greater than its square root.
e.g., 36 has the factors 1, 2, 3, 4, 6, 9, 12, 18 and 36 and √36 = 6
1,2, 3 and 4 are less than 6 while 9, 12, 18 and 36 are greater than 6
∴ If the number of factors of n between 1 and √n = 5, then the number of factors between √n and n is also 5.
Apart from this, 1, √n and n are also factors of n.
Total number of factors = 5 + 5 + 3=13
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Which of the following statements is/are true?a. A perfect square always has an odd numbers of factors.b. If a number n has five factors between 1 and the square root of n (n is a perfect square), then n has 13 factors.c. 72 has 11 factors.a)(a) and (b)b)(b) and (c)c)(a) and (c)d)All the statements are truee)None of the statements is trueCorrect answer is option 'A'. Can you explain this answer?
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Which of the following statements is/are true?a. A perfect square always has an odd numbers of factors.b. If a number n has five factors between 1 and the square root of n (n is a perfect square), then n has 13 factors.c. 72 has 11 factors.a)(a) and (b)b)(b) and (c)c)(a) and (c)d)All the statements are truee)None of the statements is trueCorrect answer is option 'A'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Which of the following statements is/are true?a. A perfect square always has an odd numbers of factors.b. If a number n has five factors between 1 and the square root of n (n is a perfect square), then n has 13 factors.c. 72 has 11 factors.a)(a) and (b)b)(b) and (c)c)(a) and (c)d)All the statements are truee)None of the statements is trueCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following statements is/are true?a. A perfect square always has an odd numbers of factors.b. If a number n has five factors between 1 and the square root of n (n is a perfect square), then n has 13 factors.c. 72 has 11 factors.a)(a) and (b)b)(b) and (c)c)(a) and (c)d)All the statements are truee)None of the statements is trueCorrect answer is option 'A'. Can you explain this answer?.
Which of the following statements is/are true?a. A perfect square always has an odd numbers of factors.b. If a number n has five factors between 1 and the square root of n (n is a perfect square), then n has 13 factors.c. 72 has 11 factors.a)(a) and (b)b)(b) and (c)c)(a) and (c)d)All the statements are truee)None of the statements is trueCorrect answer is option 'A'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Which of the following statements is/are true?a. A perfect square always has an odd numbers of factors.b. If a number n has five factors between 1 and the square root of n (n is a perfect square), then n has 13 factors.c. 72 has 11 factors.a)(a) and (b)b)(b) and (c)c)(a) and (c)d)All the statements are truee)None of the statements is trueCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following statements is/are true?a. A perfect square always has an odd numbers of factors.b. If a number n has five factors between 1 and the square root of n (n is a perfect square), then n has 13 factors.c. 72 has 11 factors.a)(a) and (b)b)(b) and (c)c)(a) and (c)d)All the statements are truee)None of the statements is trueCorrect answer is option 'A'. Can you explain this answer?.
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