Analyzing Number Patterns and Analogies

This question asks us to identify a set of numbers from the given options that shares a similar pattern or relationship with the numbers in the provided set: (273, 365, 367). To solve this type of number analogy problem, we need to carefully examine the given set and look for arithmetic operations, differences, sums of digits, or other mathematical properties that relate the numbers.

Analyzing the Given Set (273, 365, 367)

Let's explore different possible patterns within the set (273, 365, 367):

• Difference between consecutive numbers:
  • \(365 - 273 = 92\)
  • \(367 - 365 = 2\)
  This pattern of differences (92, 2) doesn't immediately suggest a simple relationship we'd expect to find easily in options.
• Relationship between the numbers themselves: 365 and 367 are consecutive odd numbers. 273 is smaller than both. This structural observation might be a clue.
• Sum of digits: Let's calculate the sum of digits for each number in the set:
  • For 273: \(2 + 7 + 3 = 12\)
  • For 365: \(3 + 6 + 5 = 14\)
  • For 367: \(3 + 6 + 7 = 16\)
  The sums of digits are 12, 14, and 16. These numbers form an arithmetic progression with a common difference of 2 (\(14 - 12 = 2\), \(16 - 14 = 2\)). This seems like a strong candidate for the underlying pattern.

Based on the analysis, the most distinct and likely pattern is that the sum of the digits of the numbers in the set (273, 365, 367) forms an arithmetic progression (12, 14, 16).

Comparing Options Based on Sum of Digits

Now, let's calculate the sum of digits for each number in the given options and see which one follows the same pattern.

Option (a): (246, 257, 358)

• For 246: \(2 + 4 + 6 = 12\)
• For 257: \(2 + 5 + 7 = 14\)
• For 358: \(3 + 5 + 8 = 16\)

The sums of digits for option (a) are 12, 14, and 16. This is exactly the same pattern and values as the given set (273, 365, 367).

Option (b): (145, 235, 325)

• For 145: \(1 + 4 + 5 = 10\)
• For 235: \(2 + 3 + 5 = 10\)
• For 325: \(3 + 2 + 5 = 10\)

The sums of digits are 10, 10, and 10. This set has all sums equal, which is not the pattern found in the given set.

Option (c): (143, 253, 246)

• For 143: \(1 + 4 + 3 = 8\)
• For 253: \(2 + 5 + 3 = 10\)
• For 246: \(2 + 4 + 6 = 12\)

The sums of digits are 8, 10, and 12. These numbers form an arithmetic progression with a common difference of 2 (\(10 - 8 = 2\), \(12 - 10 = 2\)). While this is an arithmetic progression, the actual values (8, 10, 12) are different from the given set's sums (12, 14, 16). The question asks for a set that is "similar", and option (a) provides an exact match for both the pattern and the values of the sum of digits sequence.

Option (d): (233, 343, 345)

• For 233: \(2 + 3 + 3 = 8\)
• For 343: \(3 + 4 + 3 = 10\)
• For 345: \(3 + 4 + 5 = 12\)

The sums of digits are 8, 10, and 12. This is the same pattern and values as option (c), and thus not the same as the given set (12, 14, 16).

Conclusion

Comparing the sums of digits, option (a) (246, 257, 358) yields sums 12, 14, and 16, which exactly matches the pattern and values of the sums of digits from the given set (273, 365, 367). Options (c) and (d) show a similar *type* of pattern (arithmetic progression of sums of digits) but with different values (8, 10, 12). Option (b) shows a different pattern entirely.

Therefore, the set most similar to (273, 365, 367) based on the sum of digits pattern is (246, 257, 358).

Revision Table: Number Patterns

Additional Information: Types of Number Patterns

Number pattern and analogy questions can involve various types of relationships between numbers. Recognizing these common types can help solve problems faster.

• Arithmetic Operations: Addition, subtraction, multiplication, division between consecutive numbers or numbers in specific positions.
• Differences/Ratios: Looking at the differences or ratios between numbers in the set. These differences or ratios might form a pattern themselves (like an AP or GP).
• Sum/Product of Digits: Analyzing the sum or product of the digits of each number, as seen in this problem.
• Properties of Numbers: Checking if numbers are prime, composite, perfect squares, perfect cubes, odd, even, etc.
• Positional Value of Digits: Sometimes the pattern relates to the specific digits at the units, tens, or hundreds place.
• Combinations of Patterns: Some problems might involve more than one pattern combined.

Practicing different types of number pattern questions helps improve your ability to quickly identify the relationship between numbers.