Problem 1

Let

p

and

q

be positive integers such that

\dfrac pq=1-\dfrac12+\dfrac13-\dfrac14+\cdots-\dfrac1{1318}+\dfrac1{1319}

. Prove that

p

is divisible by

1979

Step 3 of 4: Extract the prime factor

Previous step Next step

\dfrac1{660+i}+\dfrac1{1319-i}=\dfrac{1979}{(660+i)(1319-i)}

Detailed analysis

For every pair, adding the two fractions gives 1979 divided by the product of the two denominators. Thus S is 1979 times a rational number whose displayed denominator is a product of positive integers smaller than 1979.