In the following number analogy, 17 is related to 493 in a particular way. Using the same pattern, which number should complete the analogy 17 : 493 :: ? : 551?

Introduction / Context:
This problem involves a number analogy where each given number is related to another by multiplication with a constant factor. The pair 17 : 493 suggests that 493 is obtained by multiplying 17 by a specific number. Once we determine this constant multiplier, we can use it to determine which number, when multiplied by the same factor, yields 551. This checks your ability to reverse engineer a simple linear relationship from the result back to the original input.

Given Data / Assumptions:
First pair: 17 : 493. Second pair: ? : 551. All numbers are positive integers. The transformation is expected to be a simple multiplication by a fixed constant.

Concept / Approach:
We begin by checking whether 493 is divisible by 17. If it is, the quotient will be the constant that multiplies the first number to give the second. Then, to preserve the analogy, the unknown number should be 551 divided by this same constant. If 551 divides cleanly by that constant, the quotient is our missing number. This approach avoids guesswork and uses straightforward division to recover the pattern.

Step-by-Step Solution:
Step 1: Compute 493 / 17. 17 * 20 = 340, 17 * 25 = 425, 17 * 29 = 493. So 493 = 17 * 29, and the multiplier is 29. Step 2: Interpret the rule. In the first pair, the second number is the first number multiplied by 29. Step 3: Apply the same rule to the second pair. We require some number x such that x * 29 = 551. Step 4: Compute 551 / 29. 29 * 20 = 580, which is too large. Subtract one group of 29: 580 - 29 = 551. Therefore, 551 = 19 * 29. So the missing number is 19.

Verification / Alternative check:
We can summarise the pattern as output = input * 29. For the first pair, 17 * 29 = 493, which matches the given number. For the second pair, 19 * 29 = 551, which matches the second number in the analogy. The same multiplier of 29 works for both pairs, which confirms that the pattern is consistent. None of the other options, when multiplied by 29, produce 551, so 19 is uniquely determined.

Why Other Options Are Wrong:
13 * 29, 21 * 29, 23 * 29, and 11 * 29 do not equal 551; they yield different products such as 377, 609, 667, and 319 respectively. Because the problem clearly uses a constant multiplier, any number that does not restore 551 through multiplication by 29 breaks the analogy.

Common Pitfalls:
Some learners may try to decompose 493 and 551 using squares or cubes instead of checking for a simple product with a moderate sized integer. Others might mistakenly divide 551 by 17 or another candidate rather than first confirming the correct multiplier from the 17 : 493 pair. The best approach is systematic: find the factor that links the first pair, then use it as the divisor to recover the unknown number from the second pair. This process is efficient and reliable for similar number analogies.

Final Answer:
The number that correctly completes the analogy is 19.