MathDB

17

Part of 2022 AMC 10

Problems(2)

Repeating Decimals Timesink

Source: 2022 AMC 10A #17

11/11/2022
How many three-digit positive integers a‾\underline{a} b‾\underline{b} c‾\underline{c} are there whose nonzero digits aa, bb, and cc satisfy
0.a‾ b‾ c‾‾=13(0.a‾+0.b‾+0.c‾)?0.\overline{\underline{a}~\underline{b}~\underline{c}} = \frac{1}{3} (0.\overline{a} + 0.\overline{b} + 0.\overline{c})? (The bar indicates repetition, thus 0.a‾ b‾ c‾‾0.\overline{\underline{a}~\underline{b}~\underline{c}} in the infinite repeating decimal 0.a‾ b‾ c‾ a‾ b‾ c‾ ⋯0.\underline{a}~\underline{b}~\underline{c}~\underline{a}~\underline{b}~\underline{c}~\cdots)
<spanclass=′latex−bold′>(A)</span>9<spanclass=′latex−bold′>(B)</span>10<spanclass=′latex−bold′>(C)</span>11<spanclass=′latex−bold′>(D)</span>13<spanclass=′latex−bold′>(E)</span>14<span class='latex-bold'>(A) </span>9\qquad<span class='latex-bold'>(B) </span>10\qquad<span class='latex-bold'>(C) </span>11\qquad<span class='latex-bold'>(D) </span>13\qquad<span class='latex-bold'>(E) </span>14
AMCAMC 102022 AMC 10a2022 AMCnumber theory
AMC12B #15/10B #17

Source: 2022 AMC 10B #17 / 2022 AMC 12B #15

11/17/2022
One of the following numbers is not divisible by any prime number less than 10. Which is it? (A) 2606−1  2^{606} - 1 \ \ (B) 2606+1  2^{606} + 1 \ \ (C) 2607−1  2^{607} - 1 \ \ (D) 2607+1  2^{607} + 1 \ \ (E) 2607+3607  2^{607} + 3^{607} \ \
AMCAMC 122022 AMC 10B2022 AMC 12B2022 AMCAMC 10