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Try this beautiful problem from the American Invitational Mathematics Examination, AIME 2015 based on Number Theory and Series.

## Number Theory and Series – AIME 2015

The expressions A = \((1 \times 2)+(3 \times 4)+….+(35 \times 36)+37\) and B = \(1+(2 \times 3)+(4 \times 5)+….+(36 \times 37)\) are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers A and B.

- is 107
- is 648
- is 840
- cannot be determined from the given information

**Key Concepts**

**S**eries

Theory of Equations

Number Theory

## Check the Answer

But try the problem first…

Answer: is 648.

Source

Suggested Reading

AIME, 2015, Question 1

Elementary Number Theory by David Burton

## Try with Hints

First hint

B-A=\(-36+(2 \times 3)+….+(2 \times 36)\)

Second Hint

=\(-36+4 \times (1+2+3+….+18)\)

Final Step

=\(-36+(4 \times \frac{18 \times 19}{2})\)=648.

## Other useful links

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