# GATE | GATE-CS-2005 | Question 44

• Difficulty Level : Easy
• Last Updated : 28 Jun, 2021

What is the minimum number of ordered pairs of non-negative numbers that should be chosen to ensure that there are two pairs (a, b) and (c, d) in the chosen set such that “a ≡ c mod 3” and “b ≡ d mod 5”
(A) 4
(B) 6
(C) 16
(D) 24

Explanation:
a = c mod 3 (given)
Thus, ‘a’ can be any one of these values : 0, 1, 2

b = d mod 5 (given)
Thus, ‘b’ can be any one of these values : 0, 1, 2, 3, 4

Thus, ordered pair for (a, b) are :
(0, 0), (0, 1), (0, 2), (0, 3), (0, 4), (1, 0), (1, 1), (1, 2), (1, 3), (1, 4), (2, 0), (2, 1), (2, 2), (2, 3), (2, 4)

Therefore, ordered pair (a, b) has 15 combinations and ordered pair (c, d) has 1 combination.
Total combinations = 15 + 1 = 16

Hence, option (C) is correct.

Please comment below if you find anything wrong in the above post.

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