Understanding the Number Analogy Problem

The question asks us to find the relationship between the first pair of numbers (1375 and 64) and use that same relationship to find the missing number in the second pair (13579 and ?).

The given analogy is:

\(1375 : 64 \Proportion 13579 : ?\)

We need to discover how 1375 is related to 64.

Analyzing the First Pair: 1375 and 64

Let's look closely at the numbers 1375 and 64 to find a pattern. 1375 is a four-digit number, and 64 is a relatively smaller number. 64 is a perfect cube (\(4^3\)) and also a perfect square (\(8^2\)).

Let's examine the digits of 1375: 1, 3, 7, and 5.

• Sum of digits: \(1 + 3 + 7 + 5 = 16\). This doesn't directly lead to 64 easily (\(16 \times 4 = 64\)? Maybe, but let's look for a more direct link).
• Product of digits: \(1 \times 3 \times 7 \times 5 = 105\). Not 64.
• Number of digits: There are 4 digits. Could this relate to 64? We know \(4^3 = 64\). This seems promising.

Let's consider another property of the digits: whether they are odd or even.

The digits in 1375 are 1, 3, 7, and 5. All of these digits are odd digits.

The number of odd digits in 1375 is 4.

If we cube the number of odd digits, we get \(4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64\). This matches the second number in the first pair.

So, the likely relationship is: Count the number of odd digits in the first number and cube this count to get the second number.

Applying the Pattern to the Second Pair: 13579 and ?

Now let's apply this discovered pattern to the second pair, 13579 and ?. We need to find the number related to 13579 using the same rule.

Let's examine the digits of 13579: 1, 3, 5, 7, and 9.

• All of these digits (1, 3, 5, 7, 9) are odd digits.
• The number of odd digits in 13579 is 5.

According to the pattern, we need to cube the number of odd digits (which is 5) to find the missing number.

Calculation:

\(5^3 = 5 \times 5 \times 5 = 25 \times 5 = 125\)

Therefore, the missing number in the analogy is 125.

Verifying the Solution

Let's quickly check the pattern for both pairs:

• For 1375: Digits are 1, 3, 7, 5 (all odd). Number of odd digits = 4. \(4^3 = 64\). Correct.
• For 13579: Digits are 1, 3, 5, 7, 9 (all odd). Number of odd digits = 5. \(5^3 = 125\). Correct.

The pattern holds true for both pairs, and the calculated missing number is 125.

Revision Table: Number Analogy Logic

Additional Information: Exploring Number Patterns

Number analogy and series problems often test your ability to find patterns based on various mathematical operations or properties of the digits themselves. Common patterns include:

• Arithmetic Progression: Adding or subtracting a constant number.
• Geometric Progression: Multiplying or dividing by a constant number.
• Squaring or Cubing: Relating numbers to squares or cubes.
• Operations on Digits: Sum of digits, product of digits, counting odd/even digits, difference between digits, etc.
• Mixed Operations: A combination of different rules.
• Positional Value: Patterns based on the position of numbers or digits.

Practicing different types of number patterns helps in quickly identifying the logic during exams.