perplexus.info :: Numbers : Sawtooth Crossed Ratio Remix

Sawtooth Crossed Ratio Remix (Posted on 2023-03-30)

This puzzle is a variant on Sawtooth Crossed Ratio Resolution

Determine the value of {(2021!)/2027}
Where the operator {x} is defined as {x} = x - ⌊x⌋
Note: ⌊x⌋ denotes the floor function - that is, the greatest integer less than or equal to x.

*** A computer program will only be trivially different from one used in Sawtooth Crossed Ratio Resolution, so I request an analytic solution.

Submitted by Brian Smith

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Solution: (Hide)

2027 is prime so we effective want to calculate 2021! mod 2027, and that value over 2027 will be {(2021!)/2027}.

Wilson's theorem tells us that 2026! mod 2027 = 2026, from which we have 2025! mod 2027 = 1. But that is four more terms than 2021!
To rectify that we will multiply 2021! by 2022 * 2023 * 2024 * 2025 and their modular inverses 1/2022 * 1/2023 * 1/2024 * 1/2025.

Putting this all together we want to find 1/2022 * 1/2023 * 1/2024 * 1/2025 * 2025! mod 2027 = 1/2022 * 1/2023 * 1/2024 * 1/2025 mod 2027.
2022, 2023, 2024, 2025 can be replaced by their complement, yielding 1/-5 * 1/-4 * 1/-3 * 1/-2 mod 2027. Then canceling the negatives gives 1/5 * 1/4 * 1/3 * 1/2 mod 2027.

2027+1 = 2028 = 1014*2 = 676*3 = 507*4. Then 1/2 mod 2027 = 1014, 1/3 mod 2027 = 676 and 1/4 mod 2027 = 507
2027*2+1 = 4055 = 811*5. Then 1/5 mod 2027 = 811
Substituting these gets us down to 1014*676*507*811 mod 2027. This is just simple arithmetic (just larger than your common calculator): 1014*676*507*811 = 281847031128 = 139046389*2027 + 625.

So finally, 2021! mod 2027 = 625 tells us that {(2021!)/2027} = 625/2027.

Comments: ( You must be logged in to post comments.)

	Subject	Author	Date
	Hard way then easier way solution	Jer	2023-03-30 12:54:09

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