In this number analogy, “60 is to 36 as 100 is to ______”. Choose the number that completes the pair using the same numerical relationship seen in 60 and 36.

Introduction / Context:
This analogy relies on recognising how the right hand number is derived from the left hand number by using its digits. For the pair 60 : 36, we can interpret 60 as a product of two integers and then see how 36 is formed from those same digits or factors. We must apply the same construction to 100 and select the appropriate result from the options.

Given Data / Assumptions:

• First pair: 60 ↔ 36. • Second pair: 100 ↔ ? • Options: 100, 10000, 516, 1000. • We notice that 60 can be viewed as 6 × 10, suggesting a pattern involving the digits 6 and 0.

Concept / Approach:
A neat way to interpret the pair is: 60 is 6 × 10, and 36 is 6². In other words, the first number can be seen as (a × b) while the second is a², where a is one of the digits or factors. If this rule is applied to 100, which can be written as 10 × 10, then the corresponding right hand number should be 10², that is, 100. This preserves the factor-square relationship from the first pair.

Step-by-Step Solution:
Step 1: Express 60 as a product: 60 = 6 × 10. Step 2: Observe that the right side 36 is equal to 6² (6 × 6). Step 3: Interpret the pattern as “take the left factor and square it”. Step 4: Apply this idea to 100. We can write 100 as 10 × 10, where 10 is the repeating factor. Step 5: Square the factor 10. We get 10² = 100. Step 6: So, under this rule, 100 should correspond to 100, which is available as option A.

Verification / Alternative check:
We can restate the rule: for a number that can be expressed as a × 10, the corresponding partner is a². For 60, a = 6, giving 36. For 100, a is effectively 10, giving 10² = 100. None of the other options—10000, 516, or 1000—arise naturally from this pattern. Also, 10000 would be 100², which would be consistent only if the first number were 1000 expressed differently, so it does not fit the original 60 : 36 transformation.

Why Other Options Are Wrong:
• 10000: This equals 100², not 10², so it does not follow the same exponent rule applied to 60. • 516: Not related to 100 by any simple square or factor rule consistent with the first pair. • 1000: Would be 10³, which suggests a cube, whereas the original example uses a square.

Common Pitfalls:
Some candidates search for a direct percentage pattern, noting that 36 is 60 per cent of 60, and then try to apply 60 per cent to 100, which would give 60. But 60 is not among the options. The exam question is designed to encourage factor and square recognition rather than percentages. Focusing on how 36 and 60 share the digit 6 as a base for squaring reveals the intended pattern clearly.

Final Answer:
The number that correctly completes the analogy is 100.