MathDB

25

Part of 2003 AMC 10

Problems(2)

5-Digit Number Divided by 100

Source:

2/19/2008
Let n n be a 5 5-digit number, and let q q and r r be the quotient and remainder, respectively, when n n is divided by 100 100. For how many values of n n is q \plus{} r divisible by 11 11? <spanclass=′latex−bold′>(A)</span> 8180<spanclass=′latex−bold′>(B)</span> 8181<spanclass=′latex−bold′>(C)</span> 8182<spanclass=′latex−bold′>(D)</span> 9000<spanclass=′latex−bold′>(E)</span> 9090 <span class='latex-bold'>(A)</span>\ 8180 \qquad <span class='latex-bold'>(B)</span>\ 8181 \qquad <span class='latex-bold'>(C)</span>\ 8182 \qquad <span class='latex-bold'>(D)</span>\ 9000 \qquad <span class='latex-bold'>(E)</span>\ 9090
Counting Four-Digit Numbers

Source:

1/9/2009
How many distinct four-digit numbers are divisible by 3 3 and have 23 23 as their last two digits? <spanclass=′latex−bold′>(A)</span> 27<spanclass=′latex−bold′>(B)</span> 30<spanclass=′latex−bold′>(C)</span> 33<spanclass=′latex−bold′>(D)</span> 81<spanclass=′latex−bold′>(E)</span> 90 <span class='latex-bold'>(A)</span>\ 27 \qquad <span class='latex-bold'>(B)</span>\ 30 \qquad <span class='latex-bold'>(C)</span>\ 33 \qquad <span class='latex-bold'>(D)</span>\ 81 \qquad <span class='latex-bold'>(E)</span>\ 90
modular arithmetic