Problem
Count primes with the sieve of Eratosthenes
The numbers 1 to 100 are laid out in a grid. 1 is crossed out, primes found are circled and their multiples crossed. The picture only goes as far as 35. Finish the sieve and count the primes.
Reasoning · Grade 5-1 Primes and Divisors
Your answer
Watch out It looks like every number must be sieved, but past 10 nothing new falls.
How to solve
Strategy Make a Systematic List — Testing 100 numbers one at a time for divisors would take forever, so I flip the question round: instead of hunting for primes I cross out the numbers that cannot be prime, and whatever survives is the answer. Crossing out multiples of 2, then 3, then 5, then 7 is fast because each is just skip-counting. Then I list the survivors row by row so nothing is double-counted, and add up ten small counts instead of one big one.
1STEP 1
See what the picture has already done
A crossed number has some other divisor.
crossed out = has a divisor other than 1 and itself
Grade 4 already asks whether a number is prime or composite by looking for factor pairs. The sieve just does that job for a hundred numbers at once, by working forwards from the factors instead of backwards from each number.
4.OA.B.4Change Focus Count The Complement2STEP 2
Work out where the sieve is allowed to stop
Past 10 there is nothing left to cross.
10 × 10 = 100, 11 × 11 = 121 ＞ 100
This is the factor-pair idea from Grade 4: factor pairs come in a small-and-large partnership, so if the small partner is bigger than 10 the large one is too, and the product overshoots 100.
4.OA.B.4Look For A PatternThe sieve may stop once the crossing-out primes pass the square root of 100, because anything left is already prime.
Why?
A divisor always brings a partner, so if a number had a factor above its square root it would already have one below it.
🧱Divisors come in pairsEach divisor hands over its partner for free, so divisors pair off two by two unless one is its own partner.
Why?
Those smaller factors have all been swept already, so any survivor has had every possible divisor ruled out.
🧱Rule out the rest and the last one is forcedIf every possibility but one has been ruled out, the survivor must be the answer — no further checking is needed.
3STEP 3
Finish the crossing-out from 36 to 100
From 36 to 100 only multiples up to 7 remain to cross.
49 = 7 × 7, 77 = 7 × 11, 91 = 7 × 13
Skip-counting by 2, 3, 5 and 7 is something a Grade 4 student can do quickly and accurately, which is why the sieve beats testing each number separately.
4.OA.B.4Make A Systematic List4STEP 4
List the survivors ten at a time
List the survivors ten at a time.
1-10 &: 2, 3, 5, 7 ; 11-20 &: 11, 13, 17, 19 ; 21-30 &: 23, 29 ; 31-40 &: 31, 37 ; 41-50 &: 41, 43, 47 ; 51-60 &: 53, 59 ; 61-70 &: 61, 67 ; 71-80 &: 71, 73, 79 ; 81-90 &: 83, 89 ; 91-100 &: 97
Chopping the hundred into ten tidy blocks turns one long count into ten counts of at most four, and it makes a skipped number obvious.
4.OA.B.4Make A Systematic List5STEP 5
Add up the ten blocks
The ten blocks add to 25.
4 + 4 + 2 + 2 + 3 + 2 + 2 + 3 + 2 + 1 = 25
Ten one-digit numbers add up quickly, and grouping them (4+4+2 = 10, 2+3+2 = 7, 2+3+2+1 = 8, and 10+7+8 = 25) keeps the mental arithmetic easy.
4.NBT.B.4Make A Systematic List6STEP 6
Spot-check the doubtful ones
The tricky 91 and 49 are not primes.
9 + 7 = 16, not a multiple of 3
Checking a handful of tricky numbers by hand is much faster than re-doing the whole grid, and these eight are where nearly every mistake in this problem happens.
4.OA.B.4Solve An Easier Related ProblemAnswer
25 primes
4 + 4 + 2 + 2 + 3 + 2 + 2 + 3 + 2 + 1 = 25
25 out of 100 means about one number in four up to 100 is prime, which fits what the grid looks like: the first row is crowded with primes (2, 3, 5, 7, 11, 13, 17, 19 — eight of the first twenty) and they thin out steadily, with only one (97) in the last ten. That thinning makes sense, because bigger numbers have more possible factors to trip over. The answer is a count, so it is a whole number with no units, and it must be less than 100 and more than the 11 primes already circled in the picture — 25 sits comfortably in between.
Takeaway
Do not hunt for primes — cross out every multiple of 2, 3, 5 and 7, and the 25 numbers still standing are the primes up to 100.
- See what the picture has already done
- Work out where the sieve is allowed to stop
- Finish the crossing-out from 36 to 100
- List the survivors ten at a time
- Add up the ten blocks
- Spot-check the doubtful ones