For and event i made 72 dessert plates, each containing the same number of pastries. I didn't need that many plates so i removed 12 plates, and redistributed the pastries, which meant that each remaining plate received two more pastries. How many pastries were on each plate to start with?

If we take 12 plates away, then we have 72-12 = 60 plates left. Redistributing the pastries from those 12 plates, to the other 60, leaves us with no left over pastries (we get a remainder 0). This means that 72x must also be a multiple of 60 as well.

Let's find the LCM (lowest common multiple) of 60 and 72

First find the prime factorization of each:

60 = 2*2*3*5

72 = 2*2*2*3*3

We have the unique factors: 2, 3, 5

2 shows up at most 3 times, so 2^3 is one factor of the LCM

3 shows up at most 2 times, so 3^2 is a factor of the LCM

5 only shows up one time, so 5^1 is a factor of the LCM

The LCM is 2^3*3^2*5 = 8*9*5 = 360

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Now we know that the total number of pastries must be a multiple of 360 in order to have 72x be a multiple of 60.

Divide 360 over 72 and 60 to get 360/72 = 5 and 360/60 = 6. We see that the number of pastries per plate has gone up by 1. We want it to go up by 2

Let's try the next multiple of 360

Divide 720 over 72 and 60 to get 720/72 = 10 and 720/60 = 12. The jump from 10 to 12 means we have two more pastries per plate. We found the answer.

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Summary:

720 pastries total
when there are 72 plates, each plate gets 10 pastries
when there are 60 plates, each plate gets 12 pastries