Select the odd number from the given alternatives: 24, 39, 65, and 51.

Introduction / Context:
This number based odd one out question evaluates your ability to apply simple divisibility rules. The numbers are chosen so that three of them share a property related to divisibility by 3, and one does not. Recognising such patterns quickly is an essential skill in quantitative aptitude sections of competitive exams.

Given Data / Assumptions:

• The numbers given are 24, 39, 65, and 51.
• Three numbers are divisible by 3 and one is not.
• We assume familiarity with the divisibility rule for 3: a number is divisible by 3 if the sum of its digits is divisible by 3.
• No complicated calculations are required beyond simple digit addition and checking divisibility.

Concept / Approach:
The main idea is to test each number for divisibility by 3 using the digit sum rule. This rule states that if the sum of the digits of a number is a multiple of 3, then the number itself is divisible by 3. If three numbers satisfy this property and one does not, the one that fails is the odd one out. This method is quick, reliable, and ideal for mental calculations.

Step-by-Step Solution:
Step 1: Take 24. Sum of digits is 2 + 4 = 6, which is divisible by 3, so 24 is divisible by 3. Step 2: Take 39. Sum of digits is 3 + 9 = 12, which is divisible by 3, so 39 is divisible by 3. Step 3: Take 51. Sum of digits is 5 + 1 = 6, again divisible by 3, so 51 is divisible by 3. Step 4: Take 65. Sum of digits is 6 + 5 = 11, which is not divisible by 3, so 65 is not divisible by 3. Step 5: Group 24, 39, and 51 as numbers divisible by 3. Step 6: Observe that 65 does not share this property and is therefore different from the rest. Step 7: Conclude that 65 is the odd number among the given options.

Verification / Alternative check:
You can verify the result by performing direct division. 24 divided by 3 gives 8, 39 divided by 3 gives 13, and 51 divided by 3 gives 17, all exact integers. When you divide 65 by 3, you do not get an integer result, which confirms that 65 is not divisible by 3. Both the digit sum test and direct division lead to the same conclusion that 65 is the different number.

Why Other Options Are Wrong:
24 is divisible by 3 and fits the pattern shared by 39 and 51, so it is not the odd one out.
39 is also divisible by 3 and clearly belongs with the numbers that share this property.

51 is again divisible by 3 and shares the same divisibility characteristic as 24 and 39, so it cannot be considered the odd number.

Common Pitfalls:
Some candidates may mistakenly focus on parity, noticing that 24 is even while 39, 65, and 51 are odd, but this does not give a single unique odd one out with a strong logical basis. Others might miscalculate digit sums or rush through the check without verifying their steps. To avoid errors, always apply the digit sum method carefully and, if time allows, double check the result through direct division. Consistent practice with these simple rules will help you handle many number based reasoning questions efficiently.

Final Answer:
The odd number is 65.