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GRE Quantitative Reasoning QUESTION #8072
Question 1

Given that $p$ and $q$ are different prime numbers, $r$ is the least prime number greater than $p$, and $s$ is the least prime number greater than $q$.

Compare:

Column A: $r - p$

Column B: $s - q$

  • Column A is greater
  • Column B is greater
  • The two quantities are equal
  • The relationship cannot be determined✔️
Correct Answer Explanation

The difference between a prime and the next prime varies. For example:

  • $3 - 2 = 1$ (gap of 1)
  • $5 - 3 = 2$ (gap of 2)
  • $7 - 5 = 2$ (gap of 2)
  • $11 - 7 = 4$ (gap of 4)
  • $23 - 19 = 4$ (gap of 4)

Since $p$ and $q$ are different primes and we don't know which specific primes they are, the gaps $r - p$ and $s - q$ could be equal or different.

For instance:

  • If $p = 2$ and $q = 3$: then $r = 3, s = 5$, so $r - p = 1$ and $s - q = 2$, making Column B greater
  • If $p = 3$ and $q = 2$: then $r = 5, s = 3$, so $r - p = 2$ and $s - q = 1$, making Column A greater

Since the relationship depends on which primes are chosen, it cannot be determined from the given information.