Question

Which one of the following is not a prime number?

• A. 31
• B. 61
• C. 71
• D. 91

Hint:

Always test divisibility only by primes up to \(\sqrt{n\). Use quick checks: parity for \(2\), digit sum for \(3\), last digit for \(5\), and small trial divisions (e.g., \(7,11,13\)) as needed.

Answer 1

The Correct Option is D

Step 1: Prime–checking rule (up to }\(\sqrt{n}\)\textbf{).
To test if \(n\) is prime, it is enough to check divisibility only by the prime numbers \(\le \sqrt{n}\).
Useful squares: \(5^2=25,\ 6^2=36,\ 7^2=49,\ 8^2=64,\ 9^2=81,\ 10^2=100\). Step 2: Test \(31\) \((\sqrt{31}\approx 5.56)\).
Check primes \(\le 5:\ 2,3,5\).
\(\bullet\ 31\) is odd \(\) not divisible by \(2\).
\(\bullet\) Sum of digits \(3+1=4\) \(\) not a multiple of \(3\).
\(\bullet\) Does not end with \(0\) or \(5\) \(\) not divisible by \(5\).
No divisor found \(\) \(\boxed{31\ \text{is prime}}\). Step 3: Test \(61\) \((\sqrt{61}\approx 7.8(a)\).
Check primes \(\le 7:\ 2,3,5,7\).
\(\bullet\) Odd \(\) not divisible by \(2\).
\(\bullet\) Sum of digits \(6+1=7\) \(\) not a multiple of \(3\).
\(\bullet\) Last digit \(\neq 0,5\) \(\) not divisible by \(5\).
\(\bullet\) \(61\div 7=8\) remainder \(5\) \(\) not divisible by \(7\).
No divisor found \(\) \(\boxed{61\ \text{is prime}}\). Step 4: Test \(71\) \((\sqrt{71}\approx 8.4(b)\).
Check primes \(\le 7:\ 2,3,5,7\).
\(\bullet\) Odd \(\) not divisible by \(2\).
\(\bullet\) Sum of digits \(7+1=8\) \(\) not a multiple of \(3\).
\(\bullet\) Last digit \(\neq 0,5\) \(\) not divisible by \(5\).
\(\bullet\) \(71=7\times 10 + 1\) \(\) remainder \(1\) on division by \(7\).
No divisor found \(\) \(\boxed{71\ \text{is prime}}\). Step 5: Test \(91\) \((\sqrt{91}\approx 9.5(c)\).
Check primes \(\le 9:\ 2,3,5,7\).
\(\bullet\) Odd \(\) not divisible by \(2\).
\(\bullet\) Sum of digits \(9+1=10\) \(\) not a multiple of \(3\).
\(\bullet\) Last digit \(\neq 0,5\) \(\) not divisible by \(5\).
\(\bullet\) Try \(7\): \(7\times 13 = 91\) \(\) divisible by \(7\).
Hence \(91=7\times 13\) has factors other than \(1\) and itself \(\) \(\boxed{91\ \text{is not prime}}\). \[ \boxed{\text{91 is the only non-prime (Option (d)}} \]