AMC 10 · 2010 · #23
Grade 8 number-theoryPick an answer.
Tool #9 (Solve an Easier Related Problem): the full number 90! is impossible to compute, so replace it with a much smaller question about remainders. The last two nonzero digits are N mod 100, where N is 90! with its trailing zeros removed. Tool #7 (Identify Subproblems): 100 = 4 × 25 with 4 and 25 sharing no factors, so split the one hard remainder into an easy N mod 4 and a harder N mod 25, then reassemble. Tool #5 (Look for a Pattern): mod 25, each clean block of 25 consecutive numbers leaves the same footprint, and the powers of 2 repeat in a short cycle — patterns that collapse the huge product to a few small steps.
Count and strip the trailing zeros
The trailing zeros are 21 in number.
Every trailing zero is one 2 paired with one 5, and the 5s are the scarce partner, so counting 5s counts the zeros.
5.NBT.A.2Solve An Easier Related ProblemSplit the target into a 4-clock and a 25-clock
One hundred splits into two coprime clocks.
Two coprime clocks — a 4-clock and a 25-clock — together give every value 0 to 99 a unique reading, so their two answers rebuild the last two digits.
Two clocks that share no factor together give every value in the range a single unique reading.
▸ Why?
The two readings repeat together only after their least common multiple, which spans the whole range.
▸ Why?
Each value therefore matches exactly one pair of readings, so the two answers rebuild the original.
Strip the fives block by block, mod 25
Plenty of twos remain, so the small clock reads zero.
Each clean block of 25 leaves the identical footprint -1 mod 25, so you only have to count how many blocks there are.
4.OA.C.5Look For A PatternPeel the fives off the multiples of 5
Grouping in blocks tames the big clock.
Peeling one 5 off every multiple of 5 turns the scary product into a small factorial you can finish by hand.
4.OA.B.4Identify SubproblemsDivide out the 2s using cycling powers
Cycling powers undo the extra twos.
Powers of 2 loop every 20 steps mod 25, so a giant exponent collapses to a tiny one you can read off.
8.EE.A.1Look For A PatternGlue the two clocks together
Gluing the clocks gives 12, choice (A).
Among 0–99, only 12 reads 0 on the 4-clock and 12 on the 25-clock at the same time.
6.NS.B.4Identify SubproblemsTo find the last nonzero digits of a huge factorial, strip the zeros, then track the number on a 4-clock and a 25-clock and glue the two readings back together.
- Count and strip the trailing zeros
- Split the target into a 4-clock and a 25-clock
- Strip the fives block by block, mod 25
- Peel the fives off the multiples of 5
- Divide out the 2s using cycling powers
- Glue the two clocks together