A teacher wants to distribute a fixed number of bonus points among students such that each student receives the same number of points. The total number of bonus points is given by 9x+120, where x is the number of students.
Consider the following statements:
Statement I: The number of different positive integer values of x that allow exact distribution is 12.
Statement II: x must be a factor of 120.

Select the correct answer from the options below:
(a) Only Statement I is correct.
(b) Only Statement II is correct.
(c) Both Statements I and II are correct.
(d) None of the Statements is correct.

Question

A teacher wants to distribute a fixed number of bonus points among students such that each student receives the same number of points. The total number of bonus points is given by 9x+120, where x is the number of students.
Consider the following statements:
Statement I: The number of different positive integer values of x that allow exact distribution is 12.
Statement II: x must be a factor of 120.

Select the correct answer from the options below:
(a) Only Statement I is correct.
(b) Only Statement II is correct.
(c) Both Statements I and II are correct.
(d) None of the Statements is correct.

ONLiNE UPSC · Accepted Answer

Given:
9x + 120x is an integer
So, x must divide 120.
Prime factorization of 120:
120 = 23 &times; 3 &times; 5
Number of divisors = (3+1) * (1+1) * (1+1) = 4 &times; 2 &times;2 = 16
Number of positive divisors of 120 = 16.
Hence, 16 values of x are possible.
Statement I is incorrect (should be 16, not 12).

Statement II is correct.
Hence, option b is correct.

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M Question 1	⏱ 0
A teacher wants to distribute a fixed number of bonus points among students such that each student receives the same number of points. The total number of bonus points is given by 9x+120, where x is the number of students. Consider the following statements: Statement I: The number of different positive integer values of x that allow exact distribution is 12. Statement II: x must be a factor of 120. Select the correct answer from the options below: (a) Only Statement I is correct. (b) Only Statement II is correct. (c) Both Statements I and II are correct. (d) None of the Statements is correct.