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Explore the Back-StoryTry this beautiful problem from Number theory based on divisibility from AMC 10A, 2003.

Let \(n\) be a \(5\)-digit number, and let \(q\) and \(r\) be the quotient and the remainder, respectively, when \(n\) is divided by \(100\). For how many values of \(n\) is \(q+r\) divisible by \(11\)?

- \(8180\)
- \(8181\)
- \(8182\)
- \(9190\)
- \(9000\)

Number system

Probability

divisibility

Answer: \(8181\)

AMC-10A (2003) Problem 25

Pre College Mathematics

Since \(11\) divides \(q+r\) so may say that \(11\) divides \(100 q+r\). Since \(n\) is a \(5\) digit number ...soTherefore, \(q\) can be any integer from \(100\) to \(999\) inclusive, and \(r\) can be any integer from \(0\) to \(99\) inclusive.

can you finish the problem........

Since \(n\) is a five digit number then and \(11 | 100q+r\) then \(n\) must start from \(10010\) and count up to \(99990\)

can you finish the problem........

Therefore, the number of possible values of \(n\) such that \(900 \times 9 +81 \times 1=8181\)

- https://www.cheenta.com/problem-based-on-triangle-prmo-2012-problem-7/
- https://www.youtube.com/watch?v=axSw4_SuKE8

Try this beautiful problem from Number theory based on divisibility from AMC 10A, 2003.

Let \(n\) be a \(5\)-digit number, and let \(q\) and \(r\) be the quotient and the remainder, respectively, when \(n\) is divided by \(100\). For how many values of \(n\) is \(q+r\) divisible by \(11\)?

- \(8180\)
- \(8181\)
- \(8182\)
- \(9190\)
- \(9000\)

Number system

Probability

divisibility

Answer: \(8181\)

AMC-10A (2003) Problem 25

Pre College Mathematics

Since \(11\) divides \(q+r\) so may say that \(11\) divides \(100 q+r\). Since \(n\) is a \(5\) digit number ...soTherefore, \(q\) can be any integer from \(100\) to \(999\) inclusive, and \(r\) can be any integer from \(0\) to \(99\) inclusive.

can you finish the problem........

Since \(n\) is a five digit number then and \(11 | 100q+r\) then \(n\) must start from \(10010\) and count up to \(99990\)

can you finish the problem........

Therefore, the number of possible values of \(n\) such that \(900 \times 9 +81 \times 1=8181\)

- https://www.cheenta.com/problem-based-on-triangle-prmo-2012-problem-7/
- https://www.youtube.com/watch?v=axSw4_SuKE8

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