Prove that the gregorian calendar repeats itself every 400 years.

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The calendar obviously has an integral number of years and months in 400 years. If it has an integral number of weeks, then it will repeat itself after that time. The rules of the calendar eliminate a leap year in 3 out of the four century years, so there are 97 leap years in 400 years. The number of excess days of the week in 400 years can be found by ...

(303·365) mod 7 + (97·366) mod 7 = (2·1 + 6·2) mod 7 = 14 mod 7 = 0

Thus, there are also an integral number of weeks in 400 years.

The first day of the week is the same at the start of every 400-year interval, so the calendar repeats every 400 years.

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The question is missing the image given to go along with it, corresponding to the map being created. The image is attached to this answer.

The side angle side (SAS) similarity theorem states that two triangles with congruent angles and sides with identical ratios then the two triangles are similar. We have various points on the map, Home (H), Park (P), Friends house (F) and Grocery store (G).

In this example, we know the angle at the point Home on the map, is shared between the two triangles. If these two triangles are similar, then the ratio of the distances HF/HG = HP/HB. We know all of these values except for the HB which is the distance from home to the bus stop. But if these triangles are similar, we can solve for that distance.

15/9 = 10/HB
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HB = 6 blocks.

To determine if the triangles are similar we need to know the distance from home to the bus stop, and if these are indeed similar, that distance must be 6 blocks.