Four prime numbers are arranged in ascending order. The product of first three is 385 and that of last three is 1001. The largest prime number is

### Concept & Formula When given the products of overlapping subsets of a consecutive sequence, setting up a ratio will cancel out the common terms, revealing the relationship between the non-overlapping terms. Let the primes be $a, b, c, d$. $$\frac{a \times b \times c}{b \times c \times d} = \frac{a}{d}$$ ### Step-by-Step Solution * **Set up the variables:** Let the four prime numbers in ascending order be $a, b, c,$ and $d$ ($a < b < c < d$). * **Write the equations based on given data:** Product of first three: $a \times b \times c = 385$ Product of last three: $b \times c \times d = 1001$ * **Divide the two equations:** $$\frac{a \times b \times c}{b \times c \times d} = \frac{385}{1001}$$ $$\frac{a}{d} = \frac{385}{1001}$$ * **Simplify the fraction:** Notice that 385 ends in 5, so it is divisible by 5. $385 = 5 \times 77$. Since 385 is the product of three primes and 5 is one of them, the primes are 5, 7, and 11 (as $7 \times 11 = 77$). Thus, $a = 5$, $b = 7$, $c = 11$. Now simplify the fraction by dividing numerator and denominator by 77: $$\frac{385 \div 77}{1001 \div 77} = \frac{5}{13}$$ So, $\frac{a}{d} = \frac{5}{13}$. Since $a$ and $d$ must be prime numbers, $a = 5$ and $d = 13$. * The largest prime number $d$ is 13. ### Exam Strategy & Shortcut Look at the first product: 385. It ends in 5. Since these are ascending primes, one of them must be 5, and it has to be the smallest ($a=5$). If $5 \times b \times c = 385$, then $b \times c = 77$. The next consecutive primes that multiply to 77 are 7 and 11. Now use the second product: $7 \times 11 \times d = 1001$. So $77 \times d = 1001 \Rightarrow d = 13$. ### Common Pitfall Wasting time attempting to prime factorize 1001 entirely from scratch. You should leverage the common factors ($b$ and $c$) derived from the much easier number 385 to solve for $d$ quickly. ### Final Answer **Therefore, the correct answer is 13.**