In the number analogy "77 : 84 :: 121 : ____", which number follows the same multiplication pattern?

Introduction / Context:
This number analogy examines how each number in a pair can be factored and then related to the second number using the same structure. In "77 : 84", the numbers can be expressed as products, and from these products a pattern emerges. You must uncover that pattern and apply it to 121 to identify the correct partner from the options. Factor based analogies are very common in quantitative reasoning.

Given Data / Assumptions:

• First pair: 77 is related to 84.
• Second pair: 121 is related to an unknown number.
• Options: 132, 144, 88, 212 and 130.
• We expect a pattern based on factors rather than simple addition.

Concept / Approach:
Factorise 77. It can be written as 7 * 11. Now factorise 84. The simplest suitable factorisation here is 7 * 12. Observe that both numbers share the factor 7, while the second factor increases from 11 to 12, so the second number is obtained by keeping one factor constant and increasing the other by 1. Now consider 121. It equals 11 * 11, that is 11 squared. If we mirror the earlier pattern, we keep one factor 11 and increase the other factor by 1, which changes from 11 to 12. Hence the corresponding number should be 11 * 12 = 132.

Step-by-Step Solution:

Write 77 as a product: 77 = 7 * 11. Write 84 as a product that shares a factor: 84 = 7 * 12. Identify the pattern: maintain the first factor 7 and increase the second factor from 11 to 12. Express 121 as a product: 121 = 11 * 11. Apply the same pattern by keeping one 11 fixed and increasing the other from 11 to 12, giving 11 * 12 = 132.

Verification / Alternative check:
Calculate 11 * 12. The result is 132, which appears in the options. Now test other options. 144 is 12 * 12, which would correspond to increasing both factors instead of one. 88 equals 8 * 11, where the factors do not reflect the symmetrical starting point of 121. The number 212 does not arise from a simple single factor increment, and 130 does not fit any neat continuation of the 11 * 11 pattern. Therefore only 132 mirrors the operation used in the first pair.

Why Other Options Are Wrong:
Selecting 144, 88, 212 or 130 would imply a different rule from the one that produced 84 from 77. Because the original pair clearly demonstrates a pattern of keeping one factor common and increasing the other factor by 1, any number that ignores this structure fails to preserve the analogy.

Common Pitfalls:
A frequent error is to treat 77 and 84 as results of adding a fixed number, such as 77 + 7 = 84, and then try to add the same amount to 121. However, 121 + 7 = 128, which is not an option. Another mistake is to focus only on squares and choose 144 because it is 12 squared. Always examine factorisations to see if a shared structure is present first.

Final Answer:
Using the factor pattern, 77 = 7 * 11 changes to 84 = 7 * 12, and 121 = 11 * 11 changes to 132 = 11 * 12. So the analogy is "77 : 84 :: 121 : 132".