Understanding Number Pairs and Finding the Odd One Out

This question asks us to examine four pairs of numbers and identify which pair follows a different logic or rule compared to the other three. The crucial instruction is that operations should be performed on the whole numbers as given, not on their individual digits.

We are given the following number pairs:

• 64 – 4032
• 38 – 1444
• 42 – 1764
• 52 – 2704

We need to find the relationship between the number on the left side of the dash (let's call it A) and the number on the right side (let's call it B) for each pair.

Analyzing the Number Pairs and Identifying the Logic

Let's analyze each pair individually to see if we can find a consistent mathematical operation relating the left number to the right number.

Pair 1: 64 – 4032

Let A = 64 and B = 4032.

Let's consider common operations like squaring, cubing, multiplication, or simple arithmetic. Squaring the left number is often a good starting point in such problems.

Calculate the square of the left number (64):

\(64^2 = 64 \times 64\)

Let's perform the multiplication:

\(64 \times 64 = (60 + 4) \times (60 + 4) = 60 \times 60 + 60 \times 4 + 4 \times 60 + 4 \times 4\)

\(= 3600 + 240 + 240 + 16\)

\(= 3600 + 480 + 16\)

\(= 4080 + 16 = 4096\)

So, \(64^2 = 4096\). The right number is 4032.

Let's see the difference between the square and the right number:

\(4096 - 4032 = 64\)

Interestingly, the difference is equal to the original left number (64). This suggests a possible relationship: \(B = A^2 - A\).

For Pair 1: \(4032 = 64^2 - 64\).

Pair 2: 38 – 1444

Let A = 38 and B = 1444.

Let's calculate the square of the left number (38):

\(38^2 = 38 \times 38\)

\(38 \times 38 = (40 - 2) \times (40 - 2) = 40 \times 40 - 40 \times 2 - 2 \times 40 + 2 \times 2\)

\(= 1600 - 80 - 80 + 4\)

\(= 1600 - 160 + 4\)

\(= 1440 + 4 = 1444\)

So, \(38^2 = 1444\). This exactly matches the right number (B). The relationship for this pair appears to be \(B = A^2\).

Pair 3: 42 – 1764

Let A = 42 and B = 1764.

Calculate the square of the left number (42):

\(42^2 = 42 \times 42\)

\(42 \times 42 = (40 + 2) \times (40 + 2) = 40 \times 40 + 40 \times 2 + 2 \times 40 + 2 \times 2\)

\(= 1600 + 80 + 80 + 4\)

\(= 1600 + 160 + 4\)

\(= 1760 + 4 = 1764\)

So, \(42^2 = 1764\). This exactly matches the right number (B). The relationship for this pair is also \(B = A^2\).

Pair 4: 52 – 2704

Let A = 52 and B = 2704.

Calculate the square of the left number (52):

\(52^2 = 52 \times 52\)

\(52 \times 52 = (50 + 2) \times (50 + 2) = 50 \times 50 + 50 \times 2 + 2 \times 50 + 2 \times 2\)

\(= 2500 + 100 + 100 + 4\)

\(= 2500 + 200 + 4\)

\(= 2700 + 4 = 2704\)

So, \(52^2 = 2704\). This exactly matches the right number (B). The relationship for this pair is also \(B = A^2\).

Conclusion: Identifying the Odd Pair

We have found the following relationships:

• Pair 1 (64 – 4032): \(B = A^2 - A\) (\(4032 = 64^2 - 64\))
• Pair 2 (38 – 1444): \(B = A^2\) (\(1444 = 38^2\))
• Pair 3 (42 – 1764): \(B = A^2\) (\(1764 = 42^2\))
• Pair 4 (52 – 2704): \(B = A^2\) (\(2704 = 52^2\))

Pairs 2, 3, and 4 all follow the same rule where the right number is the square of the left number (\(B = A^2\)). Pair 1 follows a different rule where the right number is the square of the left number minus the left number itself (\(B = A^2 - A\)).

Therefore, the pair that is the odd one out is 64 – 4032.

The logic \(B = A^2\) is common to three pairs, while the logic \(B = A^2 - A\) is unique to the first pair. Hence, 64 – 4032 is the odd one out.

Revision Table: Number Pair Logic

Additional Information: Number Analogy Problems

Number analogy problems, like finding the odd number pair, test your ability to identify mathematical relationships and patterns between numbers. Common relationships include:

• Squaring or Cubing: The right number is the square or cube of the left number.
• Multiplication/Division: The right number is a multiple or factor of the left number.
• Addition/Subtraction: A constant value is added to or subtracted from the left number.
• Operations on Digits: (Note: This was explicitly disallowed in the problem, but is common in other problems). Sum of digits, product of digits, etc.
• Combination of Operations: Squaring and then adding/subtracting, multiplying and then adding/subtracting, etc.

Solving these problems often involves testing common operations systematically until a pattern is found that applies to most of the given pairs. The pair that does not fit the pattern is the odd one out.