XAT 2019 Question 67
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XAT
Given, m and n are two positive integers having product less than 100
Statement 1:
mn is divisible by 6 consecutive integers. This means mn must be divisible by the LCM of these 6 integers its multiples.
Only numbers from 1 to 6 satisfy this with LCM 60. Any other set of 6 consecutive integers clearly exceeds 100 as its LCM.
Only such number satisfying the condition is 60.
60 can further be expressed as the product of two positive integers in the following ways:
1 x 60, 2 x 30, 3 x 20, 4 x 15, 5 x 12 and 6 x 10.
So, no unique values of m and n can be determined from the above.
Statement 2:
m + n is a perfect square.
From this statement alone too no unique set of solutions can be determined.
From Statements 1 and 2:
Only the pair 6 x 10 satisfies the condition.
Hence, both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.