AMC 10 · 2011 · #5
Grade 6 number-theoryPick an answer.
The word "second" is the whole difficulty: finding one qualifying number is easy, but ranking them needs the full list. Tool #14 (Extreme Principle) supplies it. Take the smallest qualifying number L and use its minimality as a lever: any qualifying number leaves a remainder on division by L, and that remainder would itself qualify while being smaller than L unless it is 0. So the qualifying numbers are exactly L, 2L, 3L, …, and the second smallest is 2L. Tool #9 (Solve an Easier Related Problem) trims six divisibility conditions down to three. Tool #2 (Make a Systematic List) finds L by a short finite scan instead of a formula. Tool #7 (Identify Subproblems) splits the job into: find L, prove the list, then add digits.
Trim six conditions to three
Six conditions trim down to three.
A multiple of 4 is automatically a multiple of 2, so the smaller demands are already paid for.
A multiple of four is already a multiple of two, so the smaller demands are paid for by the larger ones.
▸ Why?
Meeting several divisibility rules at once is meeting the one rule for their least common multiple.
▸ Why?
A number divisible by four leaves no remainder on two either, so the weaker rule adds nothing.
Scan for the smallest one
The smallest such number is 60.
Only every fifth number can qualify, so a short walk through multiples of 5 settles the smallest case by hand.
6.NS.B.4Make A Systematic ListProve the list is 60, 120, 180, …
Every other one is a multiple of it.
A leftover remainder would inherit every divisibility rule while being smaller than the smallest, which is impossible, so the remainder must vanish.
4.NBT.B.6Extreme PrincipleAdd the digits of N
The digits of the second add to 3, choice (A).
Once the right number is in hand the finish is pure place value: read off the digits and add.
2.NBT.A.1Identify SubproblemsFind the smallest number that fits every rule, show the leftovers force every other one to be a multiple of it, then just count down the list: 60, then 120.
- Trim six conditions to three
- Scan for the smallest one
- Prove the list is 60, 120, 180, …
- Add the digits of N