AMC 10 · 2004 · #25
Grade 9 algebraPick an answer.
Treat the base as a variable b and find one formula that covers all 96 factors at once. The standard shift-and-subtract trick for repeating expansions turns a_b into a ratio of two polynomials. Then look at the shape of that ratio: if the numerator for base b can be rewritten as the same expression the denominator uses for base b+1, consecutive factors cancel and the whole 96-term product collapses to its two ends. Finally, minimizing n is a separate divisibility question, not a byproduct of the cancellation, so it gets its own argument.
Turn the repeating expansion into a fraction
Shifting by one period and subtracting turns the expansion into a fraction.
Shifting a repeating expansion by one whole period lines the tail up with itself, so subtracting cancels the infinite part.
Shifting the repeating expansion by one whole period lines the tail up with itself, so subtracting cancels the infinite part.
▸ Why?
Multiplying by the base to the period slides every digit along by that many places without changing any of them.
▸ Why?
The two infinite tails are identical, so the difference between the two numbers is finite and exactly computable.
Recognize the numerator as a cube minus one
The numerator is really the same cube-minus-one shape one step later.
Once the top of one factor is the bottom of the next, a long product starts eating itself.
9.A-SSE.A.2Look For A PatternMultiply and telescope
So the long product telescopes, leaving 95238 over 99 factorial.
Two copies of the same list, shifted by one, cancel everywhere except at the two ends.
9.A-SSE.A.1Look For A PatternShrink the factorial
A shared factor cancels, dropping the factorial one step to give 962.
Every factor you can peel off the top lets the factorial on the bottom lose one step.
6.NS.B.4Introduce A VariableShow 98 cannot be beaten
A divisibility check proves no smaller factorial works, so m is 962, choice (E).
Dropping one more step off the factorial would demand a factor of 98 on top, and 962 has no 7 in it.
6.NS.B.4Look For A PatternRewrite each factor so its top matches the next factor's bottom, and a 96-term product collapses to just its two ends — then check separately that nothing more can cancel.
- Turn the repeating expansion into a fraction
- Recognize the numerator as a cube minus one
- Multiply and telescope
- Shrink the factorial
- Show 98 cannot be beaten