MathLabs

Problem 1

Let pp and qq be positive integers such that pq=1−12+13−14+⋯−11318+11319\dfrac pq=1-\dfrac12+\dfrac13-\dfrac14+\cdots-\dfrac1{1318}+\dfrac1{1319}. Prove that pp is divisible by 19791979.
Step 4 of 4: Reduce the fraction
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1979 is prime1979\text{ is prime}
Detailed analysis

Since 1979 is prime and none of the denominator factors is divisible by it, reducing the rational number cannot cancel the factor 1979 in the numerator. Writing S=p/q in lowest terms therefore forces 1979 to divide p.