2021 AMC 10 A Fall

Complete problem set with solutions and individual problem pages

Problem 5 Easy

The six-digit number \underline{2} \underline{0} \underline{2} \underline{1} \underline{0} \underline{A} is prime for only one digit A. What is A ? (2021 AMC Fall 10A, Question #5)

  • A.

    1

  • B.

    3

  • C.

    5

  • D.

    7

  • E.

    9

Answer:E

Solution 1:

First, modulo 2 or 5, \underline{20210 A} \equiv A. Hence, A \neq 0,2,4,5,6,8. Second modulo 3, \underline{20210 A} \equiv 2+0+2+1+0+A \equiv 5+A. Hence, A \neq 1,4,7. Third, modulo 11, \underline{20210 A} \equiv A+1+0-0-2-2 \equiv A-3. Hence, A \neq 3. Therefore, the answer is \mathbf{( E ) 9}.

Solution 2:

202100 \Longrightarrow divisible by 2 . 202101 \Longrightarrow divisible by 3 . 202102 \Longrightarrow divisible by 2 . 202103 \Longrightarrow divisible by 11 . 202104 \Longrightarrow divisible by 2 . 202105 \Longrightarrow divisible by 5 . 202106 \Longrightarrow divisible by 2 . 202107 \Longrightarrow divisible by 3 . 202108 \Longrightarrow divisible by 2 . This leaves only A= (E) 9 .

Related Problems
Problem 3 2021 AMC 10 A Fall
Problem 4 2021 AMC 10 A Fall
Problem 6 2021 AMC 10 A Fall
Problem 7 2021 AMC 10 A Fall
Think Academy Powered by ChatGPT