Understanding the Number Analogy

This question asks us to find a missing number in a sequence based on a relationship established between the first pair of numbers. We need to identify the logical connection between 77 and 11 and then apply the exact same connection to 259 to find the unknown number.

Analyzing the Relationship: 77 : 11

Let's examine the relationship between the first number, 77, and the second number, 11.

• Division: We can check if 77 is divisible by 11.

  Calculation: $77 \div 11 = 7$. This tells us that 11 is 77 divided by 7.

• Factorization: We can find the prime factors of 77.

  Calculation: $77 = 7 \times 11$. Here, 11 is one of the prime factors.

The most direct and common relationship in such analogies is often a simple arithmetic operation. The relationship where 11 is obtained by dividing 77 by 7 (i.e., $77 / 7 = 11$) seems plausible and straightforward.

Applying the Relationship to 259

Now, we apply the identified relationship (dividing the first number by 7) to the second number, 259, to find the missing number represented by '?'.

We need to calculate:

$$ 259 \div 7 = ? $$

Let's perform the division:

$$ \frac{259}{7} $$

We can break this down:

• $210 \div 7 = 30$
• $49 \div 7 = 7$
• Adding these results: $30 + 7 = 37$.

Alternatively, using long division:

• $25 \div 7 = 3$ with a remainder of $4$ ($25 - 21 = 4$).
• Bring down the 9, making it 49.
• $49 \div 7 = 7$.
• So, $259 \div 7 = 37$.

Therefore, the missing number is 37.

Conclusion

The relationship between 77 and 11 is that 77 divided by 7 equals 11. Applying this same rule, 259 divided by 7 equals 37. The option that matches this result is 37.