AMC 10 · 2014 · #23
Grade 8 number-theorypatternPick an answer.
Dividing 1 by 9801 by hand for two hundred digits is not a plan. The way in is 1/9²=1/81, the same problem with one-digit chunks instead of two: it expands as 0.012345679, counting 0,1,2,… and skipping one value near the end. That suggests the two-digit answer, 00,01,02,…,97,99 with 98 skipped. A guess is not a solution, though, and this is exactly where the naive counting argument leaks: writing the numbers 0,1,2,3,… into two-digit slots breaks down once the numbers reach 100 and no longer fit, and the whole answer depends on where the resulting carries land. So the guess is turned into an exact claim and checked by algebra: a block B of length n works precisely when 9801 · B=10ⁿ-1, and that single equation can be verified by one shift-and-subtract, with no infinite series and no hand-waved carrying. Finally the digit total itself is what proves the block cannot be shortened, so the period question is answered rather than assumed.
Turn the block into one equation
A repeating block is one clean equation.
Sliding the decimal point one whole period leaves the same tail, so the block is trapped in a single equation.
Sliding the decimal point one whole period leaves the same tail, so the block is trapped in a single equation.
▸ Why?
Multiplying by a power of ten slides every digit along by that many places without changing any of them.
▸ Why?
The two endless tails are identical, so subtracting leaves a finite number that names the block exactly.
Test the one-digit version first
A simpler version suggests the pattern.
Shrink the problem until you can divide it out by hand, then copy the shape one digit wider.
4.OA.C.5Solve An Easier Related ProblemMultiply the guess by 99
Multiplying back collapses the guess.
Reading two digits at a time turns the monster into a 99-column subtraction whose every column gives 1.
4.NBT.B.4Organize Information In More WaysMultiply by 99 once more
A second multiplication proves it exactly.
Two easy multiplications by 99 do the work of one impossible multiplication by 9801.
8.EE.A.1Convert To AlgebraAdd the digits of the block
The digits add to 883.
Counting every pair from 00 to 99 is symmetric and easy; only one chunk has to be taken back out.
3.OA.D.9Make A Systematic ListShow 198 is the shortest period
A primality argument shows the period is shortest.
If the block secretly repeated, its digit total would split into equal parts — and a prime total refuses to split.
4.OA.B.4Eliminate Possibilities1/99² just counts by twos of digits — 00,01,02,…,97, then jumps to 99 — so its digits add to all of 00 through 99 (900) minus the skipped 98, which is 883.
- Turn the block into one equation
- Test the one-digit version first
- Multiply the guess by 99
- Multiply by 99 once more
- Add the digits of the block
- Show 198 is the shortest period