CAT 2022 Quant was dominated by Arithmetic followed by Algebra. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.
Question 4 : For natural numbers \(x, y\), and \(z\), if \(x y+y z=19\) and \(y z+x z=51\), then the minimum possible value of \(x y z\) is
xy + yz = 19
y(x + z) = 19
Observe that 19 is prime and can’t be factorized further. Since x, y and z are all natural numbers, the value of (x + z) can’t be 1. (It should at least be 2).
Therefore, y = 1 and ( x + z) = 19
yz + xz = 51
z(y + x) = 51
z(x + 1) = 51
51 = 17 * 3
Case 1: z = 17, x + 1 = 3
x = 2, y = 1, z = 17
xyz = 2 * 1 * 17 = 34
Case 2: z = 3, x + 1 = 17
x = 16, y = 1, z = 3
xyz = 16 * 1 * 3 = 48
Hence the minimum value of x * y * z = 34
The question is " For natural numbers \(x, y\), and \(z\), if \(x y+y z=19\) and \(y z+x z=51\), then the minimum possible value of \(x y z\) is "
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