Question:
\(a,b,c\) are prime numbers, \(x\) is an even number, \(y\) is an odd number. Which of the following is/are never true?
I. \(a + x = b\)
II. \(b + y = c\)
III. \(ab = c\)
IV. \(a + b = c\)
\(a,b,c\) are prime numbers, \(x\) is an even number, \(y\) is an odd number. Which of the following is/are never true?
I. \(a + x = b\)
II. \(b + y = c\)
III. \(ab = c\)
IV. \(a + b = c\)
Show Hint
Product of two primes is always composite (unless one is 1, but 1 is not prime).
Updated On: Apr 23, 2026
- I and II
- II and III
- Only III
- III and IV
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Step 1: Formula / Definition}
\[ \text{Prime }>2 \text{ is odd} \]
Step 2: Calculation / Simplification}
I: \(a+x=b\). If \(a=2\) (even prime), \(x\) even \(\Rightarrow a+x\) even. \(b\) can be 2? No, sum>2. But \(a=3, x=2 \Rightarrow b=5\) (possible)
II: \(b+y=c\). Odd + odd = even. \(c\) must be 2 (only even prime). \(3+3=6\) not prime, but \(2+3=5\) (odd+odd=odd? Wait: \(b\) odd prime, \(y\) odd \(\Rightarrow b+y\) even \(\Rightarrow c=2\) impossible since \(b,y \geq 3\)) Never true.
III: \(ab=c\). Product of two primes is composite (has factors 1,a,b,ab). Never prime. Never true.
IV: \(a+b=c\). \(2+3=5\) (possible).
\(\therefore\) III is definitely never true. II is also never true. Check options: Only III is (C). But wait: II: \(b=2\) (even prime), \(y=3 \Rightarrow c=5\) (possible). So II can be true.
Only III is never true.
Step 3: Final Answer
\[ \text{Only III} \]
\[ \text{Prime }>2 \text{ is odd} \]
Step 2: Calculation / Simplification}
I: \(a+x=b\). If \(a=2\) (even prime), \(x\) even \(\Rightarrow a+x\) even. \(b\) can be 2? No, sum>2. But \(a=3, x=2 \Rightarrow b=5\) (possible)
II: \(b+y=c\). Odd + odd = even. \(c\) must be 2 (only even prime). \(3+3=6\) not prime, but \(2+3=5\) (odd+odd=odd? Wait: \(b\) odd prime, \(y\) odd \(\Rightarrow b+y\) even \(\Rightarrow c=2\) impossible since \(b,y \geq 3\)) Never true.
III: \(ab=c\). Product of two primes is composite (has factors 1,a,b,ab). Never prime. Never true.
IV: \(a+b=c\). \(2+3=5\) (possible).
\(\therefore\) III is definitely never true. II is also never true. Check options: Only III is (C). But wait: II: \(b=2\) (even prime), \(y=3 \Rightarrow c=5\) (possible). So II can be true.
Only III is never true.
Step 3: Final Answer
\[ \text{Only III} \]
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