Which weekday does the 40th run land on?

Question

A batch job runs every 9 days, and it ran today, a Monday. Solve in your head. What day of the week does its 40th future run fall on? Aim: under 15 seconds.

Accepted Answer

The 40th future run is 40 	imes 9 = 360 days from now, so this is a "360 days later" question. Reduce 360 mod 7. The nearest multiple of 7 below is 357 = 7 	imes 51: 360 = 357 + 3 \quad\Longrightarrow\quad 360 \equiv 3 \pmod 7. Count 3 forward from Monday: Tuesday, Wednesday, Thursday. A neat shortcut: reduce before multiplying. Since 9 \equiv 2 \pmod 7, each run advances the weekday by 2, so 40 runs advance by 40 	imes 2 = 80 \equiv 80 - 77 = 3 \pmod 7. Same 3, and the smaller numbers are easier to hold in your head. Takeaway. Repeating-interval questions are "count times interval, then r…

Which weekday does the 40th run land on?

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