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# Ewa's Eggs

##### Age 11 to 14 Challenge Level:

Well done all of you who found that Ewa had 301 eggs and Tomek should pay him £23.83

Quek from Tao Nan School, Singapore, Nicholas and Oliver from Clevedon Community School in North Somerset and Rachel from Hethersett High School in Norwich all used the lowest common multiple of 2, 3, 4, 5, and 6, namely 60 and then used the 60 times table.

This is Quek's reasoning:

Since the number of eggs can be divided by 2, 3, 4, 5 and 6 and always have one left over, the number of eggs is one more than some multiple of the lowest common multiple (LCM) of 2, 3, 4, 5 and 6. This number must also be divisible by 7. The LCM of 2, 3, 4, 5 and 6 is 60. Since 60+1 is not a multiple of 7, it is not the correct answer. Working the way up the 60 times table to find a number which, plus one, is a multiple of 7, the first such number is 300, and so the number of eggs must be 301.

60 * 2 + 1=121
60 * 3 + 1=181
60 * 4 + 1=241
60 * 5 + 1=301

301 * 95/12 p = 2382p + 11/12 p
Tomek should offer to pay £23.83.

Amy of Hethersett High School, Norwich, used a similar method to this one from Emma and Elizabeth of Ipswich High School:

We wrote out the multiples of 7. We realized that if it was divisible by 5 with a remainder of 1 then it must end in 6 or 1. We also realized that if it was divisible by 2 with a remainder of 1 it must end in a 1. We went through the multiples of 7 and picked out those ending in 1 and checked them for divisiblity by 3 with remainder of 1 and came up with 301.

A slightly different approach, also starting with multiples of 7, was used by Laura, and Amelia of Ipswich High School:

I wrote down multiples of 7. I crossed out the multiples of 2, 3 and 5. (I didn't need to look for multiples of 4 and 6 because they had already been crossed out). I took 1 away from the numbers that were left and crossed out any that were not multiples of 5. This left me with 301.