Q52.Marks: +2.0UGC NET Paper 2: Computer Science and Application 26th June 2025 Shift 1
Let m and n are positive integers. Then
(A) If n ≠ 1, then m < mn.
(B) If k is composite, then k = mn where 1 < m , n > k
(C) If mn = 1 then m = 1 and n = 1 .
(D) If k is composite, then k = mn where 1 < m n < k
Which of the following is correct:
1.(A), (C), (D)✓ Correct
2. (B), (C), (D)
3.(A), (B)
4.(A), (B), (C)
Solution
The correct answer is:Option 1) (A), (C), (D)
Given: m and n are positive integers.
Concepts to use
Positive integers are 1, 2, 3, … (so m ≥ 1, n ≥ 1).
Composite number k > 1 is a number that can be written as a product of two integers strictly between 1 and k: there exist m, n with 1 < m < k, 1 < n < k and k = mn.
Statement (A): If n ≠ 1, then m < mn.
Since n is a positive integer and n ≠ 1, we must have n ≥ 2.
Multiplying m (which is positive) by n ≥ 2 strictly increases its value: mn ≥ 2m > m.
Therefore, (A) is true.
Statement (B): If k is composite, then k = mn where 1 < m , n > k.
This wording claims one of the factors is greater than k (“n > k”), which is impossible if mn = k with positive integers.
Example: k = 12 is composite. Valid factorizations are 12 = 3×4, 2×6. In every case, each factor is between 1 and 12 (strictly); none is > 12.
Therefore, (B) is false.
Statement (C): If mn = 1 then m = 1 and n = 1.
With positive integers, the only way a product equals 1 is when each factor is 1.
Therefore, (C) is true.
Statement (D): If k is composite, then k = mn where 1 < m < k and 1 < n < k.
This is exactly the definition of a composite number.
Example: k = 15 = 3×5 with 1 < 3,5 < 15.
Therefore, (D) is true.
Conclusion: The true statements are (A), (C), and (D). Hence the correct option is Option 1.