GMAT Quantitative Question of the Day

1. 420
2. 455
3. 525 (correct answer)
4. 560

Explanation: When you see a problem asking for a positive integer divisible by two numbers, you need to find their least common multiple (LCM). The smallest value of $n$ that's divisible by both 45 and 70 will be $\text{LCM}(45, 70)$. To find the LCM, first determine the prime factorizations: $45 = 3^2 \times 5$ and $70 = 2 \times 5 \times 7$. The LCM takes the highest power of each prime factor that appears: $\text{LCM}(45, 70) = 2^1 \times 3^2 \times 5^1 \times 7^1 = 2 \times 9 \times 5 \times 7 = 630$. Therefore, the least possible value of $n$ is 630, making $n - 105 = 630 - 105 = 525$. This confirms answer choice C. Let's examine why the other options are incorrect. Choice A (420) would mean $n = 525$. However, $525 = 3 \times 5^2 \times 7$ is not divisible by 45 since it lacks sufficient factors of 3. Choice B (455) gives $n = 560 = 2^4 \times 5 \times 7$, which isn't divisible by 45 because it has no factor of 3. Choice D (560) means $n = 665 = 5 \times 7 \times 19$, which is divisible by neither 45 nor 70. Strategy tip: When finding numbers divisible by multiple values, always calculate the LCM using prime factorization. Take the highest power of each prime factor that appears in any of the numbers. This guarantees the smallest number divisible by all given values.