Number System Question 5 - CAT 2020
Let N, x and y be positive integers such that N= x + y, 2 < x < 10 and 14 < y < 23 . If N > 25, then how many distinct values are possible for N?
Step-by-Step Explanation
Let's list the valid positive integer values for x and y based on the given inequalities:
- 2 < x < 10 ⇒ x ∈ {3,4,5,6,7,8,9}
- 14 < y < 23 ⇒ y ∈ {15,16,17,18,19,20,21,22}
We need to find the number of distinct values for the sum N = x + y, given that N > 25.
Let's find the absolute minimum and maximum limits for N:
- Nₘᵢₙ = xₘᵢₙ + yₘᵢₙ = 3 + 15 = 18
- Nₘₐₓ = xₘₐₓ + yₘₐₓ = 9 + 22 = 31
Since both x and y are sets of consecutive integers, their sum N will cover every single consecutive integer from 18 to 31 without any gaps.
We only need to count the values of N that are strictly greater than 25:
Valid values for N ∈ (26,27,28,29,30,31)
Counting these values gives exactly 6 distinct options.
💡 FundaMakers Tip:
When adding two complete sets of consecutive integers, the resulting sums will always form a continuous chain of integers. Find the absolute maximum sum (9 + 22 = 31) and simply count down to your constraint threshold to find the number of terms directly.