Finding the Different Number: Number Classification

This question asks us to identify the number that is different from the other three in a given set. This type of problem falls under number classification or finding the 'odd one out'. We need to look for a common property or rule that applies to three of the numbers, but not to the remaining one.

Analyzing the Given Numbers

The given numbers are 197, 193, 195, and 191. Let's examine some properties of these numbers to find a pattern. Common properties to check in such problems include:

• Whether the numbers are prime or composite.
• Divisibility rules (e.g., by 2, 3, 5, 10).
• Sum of digits.
• Presence of specific digits.
• Even or odd numbers.

Let's start by checking if these numbers are prime or composite.

• 191: To check if 191 is prime, we can try dividing it by prime numbers up to its square root. The square root of 191 is approximately $\sqrt{191} \approx 13.8$. We need to check prime numbers less than or equal to 13.8, which are 2, 3, 5, 7, 11, and 13.
  • 191 is not divisible by 2 (it's odd).
  • The sum of digits is $1+9+1=11$, which is not divisible by 3, so 191 is not divisible by 3.
  • 191 does not end in 0 or 5, so it's not divisible by 5.
  • $191 \div 7 = 27$ with a remainder.
  • $191 \div 11 = 17$ with a remainder.
  • $191 \div 13 = 14$ with a remainder.
• 193: The square root of 193 is approximately $\sqrt{193} \approx 13.8$. Check primes 2, 3, 5, 7, 11, 13.
  • 193 is not divisible by 2, 3, 5.
  • $193 \div 7 = 27$ with a remainder.
  • $193 \div 11 = 17$ with a remainder.
  • $193 \div 13 = 14$ with a remainder.
• 195: 195 ends with the digit 5. Numbers ending in 0 or 5 are divisible by 5.
  • $195 \div 5 = 39$.
• 197: The square root of 197 is approximately $\sqrt{197} \approx 14.0$. Check primes 2, 3, 5, 7, 11, 13.
  • 197 is not divisible by 2, 3, 5.
  • $197 \div 7 = 28$ with a remainder.
  • $197 \div 11 = 17$ with a remainder.
  • $197 \div 13 = 15$ with a remainder.

Identifying the Different Number

Based on our analysis of primality:

• 191 is a prime number.
• 193 is a prime number.
• 197 is a prime number.
• 195 is a composite number.

Three of the numbers (191, 193, 197) are prime numbers, while one number (195) is a composite number. Therefore, 195 is different from the other three.

Conclusion

The property that groups three of the numbers together is being a prime number. The number that does not share this property is 195, which is a composite number. Hence, 195 is the different number.

Revision Table: Number Properties

Additional Information: Prime and Composite Numbers

Understanding prime and composite numbers is fundamental in number theory and appears frequently in reasoning and aptitude tests.

• Prime Number: A natural number greater than 1 that has no positive divisors other than 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, ...
• Composite Number: A natural number greater than 1 that is not prime. It can be formed by multiplying two smaller positive integers. Examples: 4 (2x2), 6 (2x3), 8 (2x4), 9 (3x3), 10 (2x5), 12 (3x4), 15 (3x5), ...
• The number 1 is neither prime nor composite.
• The number 2 is the only even prime number.

To check if a number 'n' is prime, you only need to test divisibility by prime numbers up to the square root of 'n'. If 'n' is not divisible by any prime number less than or equal to $\sqrt{n}$, then 'n' is prime.