Odd Number Analysis: Divisor Count
The question asks to identify the number that does not follow the pattern shared by the others in the list: 24, 36, 72, 168.
Step 1: Prime Factorization
First, find the prime factorization for each number:
- $24 = 2^3 \times 3^1$
- $36 = 2^2 \times 3^2$
- $72 = 2^3 \times 3^2$
- $168 = 2^3 \times 3^1 \times 7^1$
Step 2: Calculate Number of Divisors
The number of divisors for a number $n = p_1^{a_1} p_2^{a_2} \dots p_k^{a_k}$ is calculated as $(a_1+1)(a_2+1)\dots(a_k+1)$. Let's calculate this for each number:
- Number of divisors for 24: $(3+1)(1+1) = 4 \times 2 = 8$ (Even)
- Number of divisors for 36: $(2+1)(2+1) = 3 \times 3 = 9$ (Odd)
- Number of divisors for 72: $(3+1)(2+1) = 4 \times 3 = 12$ (Even)
- Number of divisors for 168: $(3+1)(1+1)(1+1) = 4 \times 2 \times 2 = 16$ (Even)
Step 3: Identify the Odd Number
Observe the number of divisors calculated:
- 24 has 8 divisors (Even).
- 36 has 9 divisors (Odd).
- 72 has 12 divisors (Even).
- 168 has 16 divisors (Even).
The number 36 is the only one with an odd number of divisors. All other numbers (24, 72, 168) have an even number of divisors.
Conclusion
Based on the count of divisors, 36 is the odd number out.
Important Questions from Odd Man Out
Four letter-clusters have been given, out of which three are alike in some manner and one is different. Select the one that is different. The odd one out is not based on the number of consonants/vowels or their position in the letter cluster.
The same operation(s) are followed in all the given number pairs except one. Find that odd number pair. (NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13 – Operations on 13 such as adding/deleting/multiplying etc., to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed.)
Three of the following numbers are alike in a certain way and one is different. Pick the odd one out. (NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13 – Operations on 13 such as adding/deleting/multiplying etc., to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed.)
'डटे रहना' (Persist), 'रुकना' (Endure) से उसी प्रकार संबंधित है, जिस प्रकार 'टालना' (Stall) से __________ 'से संबंधित है।
Identify the figure given in the options which when put in place of ? will logically complete the series.
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