Four prime numbers are listed in order. If the product of the first three is 105 and the product of the last three is 385, what is the largest prime number?
11
Let the four primes in order be a, b, c, d, with \(a \times b \times c = 105\) and \(b \times c \times d = 385\).
Factorising, \(105 = 3 \times 5 \times 7\), so a, b, c are 3, 5, 7 in some order.
Factorising, \(385 = 5 \times 7 \times 11\), so b, c, d are 5, 7, 11 in some order.
The common terms b and c must be 5 and 7, which fixes \(a = 3\) and \(d = 11\).
So the four primes in order are 3, 5, 7, 11.
Hence, the largest prime number is 11.
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