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M is the smallest number that has exactly 15 factors. Which of the following is correct? A. M – 23 a perfect square B. Unit digit of M is 9 C. Sum of digits of M is a perfect square D. M is a perfect square

M is the smallest number that has exactly 15 factors. Which of the following is correct? A. M – 23 a perfect square B. Unit digit of M is 9 C. Sum of digits of M is a perfect square D. M is a perfect square Correct Answer Only A, C and D

Concept Used:

N = ap × bq × cr Where a, b and c are prime factors of N.

Number of factors of N = (p +1) × (q + 1) × (r + 1)

Calculation:

15 can be written as 1 × 15 or 3 × 5

Underlying prime factorization of M can be of ap ⇒ a14

Underlying prime factorization of M can be of ap × bq ⇒ a2 × b4

Smallest possible number in each case

a14 = 214

a2 × b4 = 32 × 24

⇒ 9 × 16

⇒144

M = 144

144 is the smallest number that has exactly 15 factors.

M – 23 = 144 – 23

⇒ 121

⇒ 112

∴ A is correct

Unit digit of M is 4

∴ B is not correct

Sum of digits of M = 1 + 4 + 4 = 9

9 = 32

∴ C is correct

√M = √144 = 12

M is a perfect square

∴ D is correct

∴ A, C and D are correct

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