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Try this beautiful problem from the American Invitational Mathematics Examination, AIME 2012 based on Numbers of positive integers.

Find the number of positive integers with three not necessarily distinct digits, \(abc\), with \(a \neq 0\) and \(c \neq 0\) such that both \(abc\) and \(cba\) are multiples of \(4\).

- is 107
- is 40
- is 840
- cannot be determined from the given information

Integers

Number Theory

Algebra

But try the problem first...

Answer: is 40.

Source

Suggested Reading

AIME, 2012, Question 1.

Elementary Number Theory by David Burton .

First hint

Here a number divisible by 4 if a units with tens place digit is divisible by 4

Second Hint

Then case 1 for 10b+a and for 10b+c gives 0(mod4) with a pair of a and c for every b

[ since abc and cba divisible by 4 only when the last two digits is divisible by 4 that is 10b+c and 10b+a is divisible by 4]

and case II 2(mod4) with a pair of a and c for every b

Then combining both cases we get for every b gives a pair of a s and a pair of c s

Final Step

So for 10 b's with 2 a's and 2 c's for every b gives \(10 \times 2 \times 2\)

Then number of ways \(10 \times 2 \times 2\) = 40 ways.

- https://www.cheenta.com/cubes-and-rectangles-math-olympiad-hanoi-2018/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s

Content

[hide]

Try this beautiful problem from the American Invitational Mathematics Examination, AIME 2012 based on Numbers of positive integers.

Find the number of positive integers with three not necessarily distinct digits, \(abc\), with \(a \neq 0\) and \(c \neq 0\) such that both \(abc\) and \(cba\) are multiples of \(4\).

- is 107
- is 40
- is 840
- cannot be determined from the given information

Integers

Number Theory

Algebra

But try the problem first...

Answer: is 40.

Source

Suggested Reading

AIME, 2012, Question 1.

Elementary Number Theory by David Burton .

First hint

Here a number divisible by 4 if a units with tens place digit is divisible by 4

Second Hint

Then case 1 for 10b+a and for 10b+c gives 0(mod4) with a pair of a and c for every b

[ since abc and cba divisible by 4 only when the last two digits is divisible by 4 that is 10b+c and 10b+a is divisible by 4]

and case II 2(mod4) with a pair of a and c for every b

Then combining both cases we get for every b gives a pair of a s and a pair of c s

Final Step

So for 10 b's with 2 a's and 2 c's for every b gives \(10 \times 2 \times 2\)

Then number of ways \(10 \times 2 \times 2\) = 40 ways.

- https://www.cheenta.com/cubes-and-rectangles-math-olympiad-hanoi-2018/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s

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