AMC 10 · 2013 · #10
Grade 8 number-theoryPick an answer.
An endless decimal is hard to reason about directly, so name it (Tool #4): let x = 0.ababab… and let N stand for the two-digit block value 10a+b. Multiplying x by 100 slides the decimal exactly one block over and leaves the same infinite tail, so subtracting kills the tail and turns the decimal into the fraction N/99. Setting that equal to 1/n gives nN = 99, which turns an infinite-decimal question into a finite factoring question. From there, list all factor pairs of 99 (Tool #2) and cross off the ones whose block has a = b (Tool #3). The listing must run in both directions: finding n ∣ 99 only says which n could work, so each survivor still has to be checked to make sure its digits really are two different digits.
Name the decimal and its block
The repeating block is that block over 99.
Shifting by exactly one period lines the endless tail up with itself, so subtracting cancels the infinite part and only the block is left.
Shifting by exactly one period lines the endless tail up with itself, so subtracting leaves only the block.
▸ Why?
Multiplying by a power of ten slides every digit along by that many places without changing any of them.
▸ Why?
The two tails are identical, so the difference between the two numbers is finite and exact.
Turn the equation into a factor pair
The equation becomes a plain factor pair.
If two whole numbers multiply to 99, each one is a divisor of 99 and the other is whatever is left over.
6.EE.B.7Introduce A VariableList every factor pair of 99
There are only six such pairs.
Once the problem says n times something equals 99, there are only six numbers left to look at.
4.OA.B.4Make A Systematic ListCross off the blocks with equal digits
The digit condition rejects half of them.
The block is a two-place slot, so a small number like 9 sits in it as 09 — the zero in front is a real digit, not padding.
5.NBT.A.3Eliminate PossibilitiesConfirm the survivors and add
The survivors add to 143, choice (D).
Showing an n must divide 99 only narrows the field; the answer is only safe after each finalist is actually run through the division.
4.NBT.B.4Make A Systematic ListAny decimal that repeats a two-digit block is that block over 99, so 1/n = 0.ab just means n times the block equals 99 — and asking for two different digits only throws out the blocks 11, 33, 99.
- Name the decimal and its block
- Turn the equation into a factor pair
- List every factor pair of 99
- Cross off the blocks with equal digits
- Confirm the survivors and add