Analyzing Numbers to Find the Odd One Out

The question asks us to identify the number that is different from the rest among the given four options: 99, 117, 135, and 105. Three of these numbers share a common characteristic or pattern, while one does not. We need to examine the properties of these numbers to find this difference.

Examining Number Properties and Patterns

Let's look at potential patterns or properties that might distinguish one number from the others. Common properties to check include:

• Divisibility by specific numbers (like 2, 3, 5, 9, 10, 11, etc.)
• Sum of digits
• Whether the numbers are prime or composite
• Relationships between the numbers (like arithmetic or geometric progressions, although less likely with just four numbers)

Checking Divisibility by 9

A common pattern in number series questions is divisibility. Let's check if these numbers are divisible by common factors. Notice that all sums of digits are relatively small (9+9=18, 1+1+7=9, 1+3+5=9, 1+0+5=6), suggesting divisibility by 3 or 9 might be relevant.

Let's test divisibility by 9. A number is divisible by 9 if the sum of its digits is divisible by 9.

• 99: Sum of digits = $9 + 9 = 18$. Since 18 is divisible by 9 ($18 \div 9 = 2$), 99 is divisible by 9. ($99 \div 9 = 11$)
• 117: Sum of digits = $1 + 1 + 7 = 9$. Since 9 is divisible by 9 ($9 \div 9 = 1$), 117 is divisible by 9. ($117 \div 9 = 13$)
• 135: Sum of digits = $1 + 3 + 5 = 9$. Since 9 is divisible by 9 ($9 \div 9 = 1$), 135 is divisible by 9. ($135 \div 9 = 15$)
• 105: Sum of digits = $1 + 0 + 5 = 6$. Since 6 is not divisible by 9, 105 is not divisible by 9.

Let's summarize the divisibility by 9 in a table:

Identifying the Odd Number based on Divisibility

Based on our analysis, we can see a clear pattern: the numbers 99, 117, and 135 are all divisible by 9. The number 105, however, is not divisible by 9.

Therefore, 105 is the number that is different from the other three.

Revision Table: Odd One Out Analysis

Additional Information: Divisibility Rules for Number Reasoning

Understanding divisibility rules is very helpful in solving odd one out number questions and other number series or numerical reasoning problems. Here are a few common ones:

• Divisibility by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).
• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
• Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
• Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.
• Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3.
• Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
• Divisibility by 10: A number is divisible by 10 if its last digit is 0.
• Divisibility by 11: A number is divisible by 11 if the alternating sum of its digits is divisible by 11 (e.g., for 121, $1 - 2 + 1 = 0$, divisible by 11).

Applying these rules systematically can help uncover the underlying pattern quickly.