Five prime numbers are arranged in ascending order. The ratio of the product of the first three of them to that of the last three of them is 35 : 323. What is the difference between the smallest and the largest numbers ?

Correct — D, (d) 14. The whole problem collapses once you notice that the middle prime cancels.

Call the five primes p1 < p2 < p3 < p4 < p5. The first three multiply to p1·p2·p3 and the last three to p3·p4·p5 — and p3 appears in BOTH products, since with five numbers the first three and the last three overlap in the third. So (p1·p2·p3) / (p3·p4·p5) = (p1·p2) / (p4·p5) = 35 / 323.

Now factorise the two sides. 35 = 5 × 7. 323 = 17 × 19 (a useful one to recognise: 323 = 324 − 1 = 18² − 1 = 17 × 19). The two have no common factor, so the fraction is already in lowest terms.

Since p1, p2, p4 and p5 are all primes, the equation p1·p2·323 = p4·p5·35 can be satisfied in only one way: by unique factorisation, the pair {p1, p2} must be {5, 7} and the pair {p4, p5} must be {17, 19}. Ascending order then fixes p1 = 5, p2 = 7, p4 = 17, p5 = 19.

The smallest number is 5 and the largest is 19, so the difference is 19 − 5 = 14.

A point worth noticing, because it shows the question was carefully built: p3 is NOT determined. It only has to be a prime strictly between 7 and 17, so it can be 11 or 13, and both give a valid set of five primes. The question asks for the difference between the smallest and the largest, which is the same either way — so an item that looks under-determined is in fact perfectly determinate for what it asks.

Verify the set before writing the answer. Take the five primes as 5, 7, 11, 17, 19: the product of the first three is 5 × 7 × 11 = 385, the product of the last three is 11 × 17 × 19 = 3,553, and 385/3,553 reduces by 11 to 35/323 — the ratio in the question. Take 13 instead of 11 and the same cancellation happens with 13. The check costs ten seconds and confirms both that the primes are right and that the middle one really is free.

Why the others are wrong

• (a)8 — Eight is the difference between two of the primes in the set, but not the two the question asks for — 13 − 5 and 19 − 11 both give 8. It is what comes of taking the undetermined middle prime as one of the endpoints. The smallest and largest of five ascending numbers are the first and the fifth, and here those are 5 and 19.
• (b)10 — Ten is 17 − 7, that is, the difference between the second and the fourth primes. This is the value produced by a candidate who correctly finds the pairs {5, 7} and {17, 19} and then subtracts the wrong member of each pair — taking the larger of the small pair and the smaller of the large pair. Having found the four primes, write them out in order before subtracting.
• (c)12 — Twelve arises from mixing the pairs the other way: 19 − 7 or 17 − 5. Both use one correct endpoint and one wrong one. It is the commonest slip in this item because the four primes 5, 7, 17 and 19 are easy to obtain and easy to muddle at the last step, when the arithmetic feels finished.

Two ideas do the work here. The first is that in a product of consecutive terms, overlapping factors cancel — with five numbers, the first three and the last three share the middle one, so the ratio reduces to a comparison of two pairs. The second is unique factorisation: every integer greater than one has exactly one factorisation into primes, so if a product of two primes equals 35, those primes can only be 5 and 7. That second principle is what makes prime-based puzzles solvable at all, and it is why such problems usually have a single answer even when they look short of data. It is worth being able to factorise the small semiprimes on sight — 91 = 7 × 13, 187 = 11 × 17, 221 = 13 × 17, 247 = 13 × 19, 289 = 17², 319 = 11 × 29, 323 = 17 × 19, 391 = 17 × 23.

Key facts

• Among five terms in ascending order, the first three and the last three share the middle term, which cancels in a ratio.
• 35 = 5 × 7 and 323 = 17 × 19; the two have no common factor.
• 323 is recognisable as 18² − 1 = 17 × 19.
• By unique factorisation, a product of two primes equal to 35 must be 5 × 7, and equal to 323 must be 17 × 19.
• The five primes are therefore 5, 7, p, 17, 19 with p equal to 11 or 13.
• The difference between the largest and the smallest is 19 − 5 = 14, whichever value p takes.
• A question can be fully determinate for the quantity it asks about while leaving other quantities free.

Common traps

• Failing to notice that the third prime is in both products and cancels.
• Subtracting the wrong pair of primes after finding all four correctly.
• Abandoning the question because the middle prime cannot be pinned down.
• Trying to search for five primes by trial instead of factorising 35 and 323.

• practice — not a real PYQ

  Four prime numbers are arranged in ascending order. The ratio of the product of the first two to the product of the last two is 15 : 187. What is the sum of the four numbers ?

  • (a)36
  • (b)38
  • (c)40
  • (d)42

  Answer(a) 36