CAT 2022

Slot 1

QA

Question & Solution

Step 1: Check possible values of y

If $y = 19$, then: $$x + z = 1$$ This is impossible because $x$ and $z$ are natural numbers (minimum sum = 1 + 1 = 2).

Therefore: $$y = 1$$ Then: $$x + z = 19$$

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Step 2: Use the second condition Given: $$z(x + y) = 51$$

Since $y = 1$, this becomes: $$z(x + 1) = 51$$

Factorising 51: $$51 = 3 \times 17$$

So the possible natural number values of $z$ are:

• $z = 3$
• $z = 17$

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Case 1: $z = 3$

From $x + z = 19$: $$x = 16$$

Product: $$xyz = 16 \cdot 1 \cdot 3 = 48$$

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Case 2: $z = 17$

From $x + z = 19$: $$x = 2$$

Product: $$xyz = 2 \cdot 1 \cdot 17 = 34$$

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Final Answer: Minimum possible value of $xyz$ is $\boxed{34}$.