This question asks us to identify the number that is different from the rest among the given options. We need to find a common property shared by three of the numbers, while the fourth number does not possess that property. Let's examine the given numbers:

• 163
• 167
• 173
• 171

We can analyze these numbers based on various properties, such as whether they are even or odd, their divisibility by other numbers, or whether they are prime or composite.

Analysing the Given Numbers

All four numbers (163, 167, 173, and 171) are odd numbers, so this property does not help us find the different one.

Let's consider their divisibility and whether they are prime or composite numbers.

Checking for Number Properties: Prime or Composite

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

A composite number is a natural number greater than 1 that is not prime. It has at least one divisor other than 1 and itself.

Examining 163:

To check if 163 is prime, we can test for divisibility by prime numbers up to the square root of 163. The square root of 163 is approximately 12.7. The prime numbers less than 12.7 are 2, 3, 5, 7, 11.

• 163 is not divisible by 2 (it's odd).
• Sum of digits $1+6+3 = 10$, which is not divisible by 3. So, 163 is not divisible by 3.
• 163 does not end in 0 or 5, so it is not divisible by 5.
• $163 \div 7 = 23$ with a remainder. Not divisible by 7.
• $163 \div 11 = 14$ with a remainder. Not divisible by 11.

Since 163 is not divisible by any prime number up to its square root, 163 is a prime number.

Examining 167:

To check if 167 is prime, we test for divisibility by prime numbers up to the square root of 167 (approximately 12.9). The primes are 2, 3, 5, 7, 11.

• 167 is not divisible by 2.
• Sum of digits $1+6+7 = 14$, not divisible by 3.
• Not divisible by 5.
• $167 \div 7 = 23$ with a remainder. Not divisible by 7.
• $167 \div 11 = 15$ with a remainder. Not divisible by 11.

Since 167 is not divisible by any prime number up to its square root, 167 is a prime number.

Examining 173:

To check if 173 is prime, we test for divisibility by prime numbers up to the square root of 173 (approximately 13.15). The primes are 2, 3, 5, 7, 11, 13.

• 173 is not divisible by 2.
• Sum of digits $1+7+3 = 11$, not divisible by 3.
• Not divisible by 5.
• $173 \div 7 = 24$ with a remainder. Not divisible by 7.
• $173 \div 11 = 15$ with a remainder. Not divisible by 11.
• $173 \div 13 = 13$ with a remainder. Not divisible by 13.

Since 173 is not divisible by any prime number up to its square root, 173 is a prime number.

Examining 171:

To check if 171 is prime or composite, we can test for divisibility.

• Sum of digits $1+7+1 = 9$. Since 9 is divisible by 3 (and 9), 171 is divisible by 3 (and 9).

We can find factors for 171:

• $171 = 3 \times 57$
• $171 = 9 \times 19$

Since 171 has factors other than 1 and 171 (like 3, 9, 19, 57), 171 is a composite number.

Identifying the Different Number

Based on our analysis:

• 163 is a prime number.
• 167 is a prime number.
• 173 is a prime number.
• 171 is a composite number.

Three of the numbers (163, 167, 173) are prime numbers, while one number (171) is a composite number. Therefore, 171 is different from the rest.

Revision Table: Number Properties

Additional Information on Number Types

Understanding the difference between prime and composite numbers is fundamental in number theory.

Prime Numbers: These are the building blocks of multiplication. Examples include 2, 3, 5, 7, 11, 13, 17, 19, 23, etc. Note that 2 is the only even prime number.

Composite Numbers: Any integer greater than 1 that is not prime is composite. Examples include 4 (divisible by 2), 6 (divisible by 2 and 3), 8 (divisible by 2 and 4), 9 (divisible by 3), 10 (divisible by 2 and 5), 12 (divisible by 2, 3, 4, 6), etc.

The number 1 is neither prime nor composite. It is considered a unique number.