Which one of the following is the average of all prime numbers between 21 and 55 ?

Correct — C, (c) 38·00. Take 'between 21 and 55' as excluding both endpoints, which costs nothing here because neither is prime anyway: 21 = 3 x 7 and 55 = 5 x 11. The primes in that stretch are 23, 29, 31, 37, 41, 43, 47 and 53 — eight of them. The two numbers that decide this question are the ones a hurried candidate lets through: 49 = 7 x 7 and 51 = 3 x 17. Both are odd, neither ends in 5, and both sit in the middle of the range where you are counting fastest. The eight primes sum to 304, and 304 divided by 8 is 38 exactly — printed here as 38·00, the booklet setting its decimal point as a raised middle dot as it does throughout this paper. If you do not trust a running total of eight numbers, regroup them from the outside in, which uses every member and so is not a shortcut but the same sum rearranged: 23 + 53 = 76, 29 + 47 = 76, 31 + 43 = 74, 37 + 41 = 78. Those four pair-sums average 76, and half of 76 is 38.

Why the others are wrong

• (a)35·85 — This is what seven of the eight primes average to. Leave 53 out — the temptation is real, because 53 sits within two of the boundary 55 and the eye stops there — and the remaining seven sum to 251, giving 251 / 7 = 35·857..., which is this option to two places. The defect is in the list, not in the division, and two cheap guards catch it. Count the primes and write the count down before you add anything, so the divisor is fixed independently of the total. Then sanity-check the result against the spread: the eight values run from 23 to 53, and a mean of 35·85 sits below 37, which is only the fourth value up — too low for a set this evenly distributed, which is the signal that a large member has been dropped.
• (b)36·71 — The same defect with a different casualty: drop 47 and the other seven sum to 257, and 257 / 7 = 36·714..., which is this option to two places. Notice what both this option and (a) have in common — a seven-term list divided by seven. That is the single commonest way this question is failed, and it is worth a deliberate habit: enumerate the primes, count them, and only then start adding. There is also a check available once you have the sum. 304 is a multiple of 8, so the mean here comes out whole; a value carrying an awkward fractional tail is evidence that either the sum or the divisor has changed, not that the arithmetic was merely untidy.
• (d)39·00 — 39 is the midpoint of 23 and 55 — the smallest prime in the range paired with the range's upper BOUND instead of with the largest prime in it. It is also what the list averages to if 31 is the member left out (273 / 7 = 39). Both routes carry the same warning about the outside-in trick: pairing the extremes is only a legitimate shortcut when every member is paired off and the pairs are taken from inside the set, which is what makes 23 + 53 the pair to use. Averaging the two endpoints of the interval is a different operation altogether, and it agrees with the true mean only when the values happen to be symmetric about their centre.

Two ideas meet here, and the arithmetic mean is the easier of them. A prime has exactly two distinct positive divisors, 1 and itself, which is why 1 is not prime; every other whole number above 1 is composite. To test a number for primality you only have to try prime divisors up to its square root, because a factor above the square root must be paired with one below it. For everything under 121 that means testing 2, 3, 5 and 7 and nothing else, since 11 x 11 = 121. Run that test across 22 to 54 and the composites fall out cleanly: the evens go on 2; 27, 33, 39, 45 and 51 go on 3; 25 and 35 go on 5; and 49 goes on 7. What is left is the eight-member list. The mean is then the sum divided by the count, and the count is a separate fact that has to be established with as much care as the sum — an average is a ratio, and it goes wrong just as easily from the denominator as from the numerator.

Key facts

• The primes strictly between 21 and 55 are 23, 29, 31, 37, 41, 43, 47 and 53 — eight numbers summing to 304, so the mean is 38.
• 49 = 7 x 7 and 51 = 3 x 17 are the two composites in this range that most often get counted as prime; both are odd and neither ends in 5.
• 21 = 3 x 7 and 55 = 5 x 11, so whether 'between' is read as inclusive or exclusive makes no difference on this item.
• For any number below 121, testing divisibility by 2, 3, 5 and 7 settles primality, because the next prime, 11, has a square of 121.
• There are 25 primes below 100 and 15 primes between 21 and 100.
• 1 is not a prime number — it has only one distinct positive divisor — and 2 is the only even prime.
• The booklet sets every decimal point in this paper as a raised middle dot, so 38·00 is the ordinary number 38 written to two places.

Common traps

• Counting 49 or 51 as prime. They are the two composites in this range that survive a careless glance.
• Fixing the divisor from memory rather than from the list you just built — dividing eight numbers by seven produces a plausible decimal and no warning.
• Reading 'between 21 and 55' as though the endpoints mattered. Here they do not, but on a range bounded by an actual prime the reading changes the answer.
• Treating the average of the two extreme values as a shortcut for the mean. It matches only when the values are symmetric about their centre.
• Adding the eight numbers in one unbroken run under time pressure instead of pairing them, which is where a single slip goes unnoticed.

• practice — not a real PYQ

  Which one of the following is the average of all prime numbers between 20 and 40 ?

  • (a)28·20
  • (b)29·40
  • (c)30·00
  • (d)31·80

  Answer(c) 30·00

• practice — not a real PYQ

  How many prime numbers lie between 40 and 70 ?

  • (a)5
  • (b)6
  • (c)7
  • (d)8

  Answer(c) 7