AMC 10 · 2002 · #15
Grade 4 number-theoryPick an answer.
AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The tempting plan is to hunt for four actual primes, but that is slow and unnecessary. Tool #16 (Change Focus) says look at what the question really asks: only the sum, and the sum of four two-digit numbers is 10×(sum of tens digits) + (sum of units digits). So the sum is fixed the moment we know which four digits land in the units place. To find those, Tool #3 (Eliminate Possibilities) rules out any digit that cannot end a prime: a two-digit prime is never even and never ends in 5. That forces the split of the eight digits into a tens group and a units group. Tool #7 (Identify Subproblems) then finishes with two easy sub-sums.
Which digits can end a prime
A two-digit prime is never even and never ends in 5, so its units digit must be one of 1, 3, 7, 9.
Any two-digit number ending in an even digit or in 5 already has 2 or 5 as a factor, so it can never be prime.
4.OA.B.4Eliminate PossibilitiesForce the tens and units groups
Only four digits can end a prime and there are exactly four units slots, so the leftovers 2, 4, 5, 6 are forced into the tens place.
With exactly four legal units digits for four units slots, the other four digits are squeezed into the tens slots, and place value fixes the sum before the primes are even built.
With the units digits forced, the remaining digits are squeezed into the tens slots, and place value does the rest.
▸ Why?
A number is its digits weighted by their places, so each digit contributes a known multiple.
▸ Why?
Whether a number can be prime is decided by its last digit for the small primes, which is what forces the split.
Add the two place-value groups
Sum = 10 × (2+4+5+6) + (1+3+7+9) = 170 + 20 = 190, which is choice (E).
Grouping all the tens together and all the units together turns four two-digit sums into one multiply-by-ten plus one small addition.
4.NBT.B.5Identify SubproblemsYou never have to find the primes: even digits and 5 can't end a prime, so 1,3,7,9 are the units and 2,4,5,6 are the tens, and the sum is 10 × 17 + 20 = 190.
- Which digits can end a prime
- Force the tens and units groups
- Add the two place-value groups