AMC 10 · 2010 · #11
Grade 7 probabilitynumber-theoryPick an answer.
Naming the two free digits turns every palindrome into one algebraic expression. That expression splits into two pieces, and checking each piece for divisibility by 7 is far easier than testing 90 numbers one by one. Counting the winners and dividing by the total then gives the probability.
Write the palindrome with digits
Two digits build the whole palindrome.
One formula with two dials, a and b, stands in for all 90 palindromes at once.
6.EE.A.2Introduce A VariableSplit the divisibility into two parts
The outer coefficient already has a factor of seven.
The a-part is a free pass for 7, so only the b-part decides the outcome.
The outer digit contributes a multiple of seven on its own, so only the inner digit decides the outcome.
▸ Why?
A palindrome is its digits weighted by their places, so each digit contributes one fixed multiplier.
▸ Why?
A part that is already a multiple of seven leaves the remainder untouched, so only the other part matters.
Find which inner digits work
So only two inner digits work.
Only the remainder of b matters, and just two digits land on a multiple of 7.
4.OA.B.4Make A Systematic ListCount winners over total
The ratio is 1/5, choice (E).
Free outer digit times 2 good inner digits, divided by all 90, gives a clean 1/5.
7.SP.C.7Change Focus Count The ComplementBecause 1001 is already a multiple of 7, only the middle digit decides the answer, and 2 of its 10 choices work, so the chance is 1/5.
- Write the palindrome with digits
- Split the divisibility into two parts
- Find which inner digits work
- Count winners over total