Ethel went to the farmers market to buy some eggs (not in a dozen cartons) on the way to the car she accidentally dropped the basket off eggs and all her eggs broke she couldn’t remember the exact number of eggs,but when she picked them out two at a time there was 1 egg left.the same happened when she picked them out 3,4,5,6 at a time but when she took them 7 at a time they come out even what’s the smallest number she could of had?
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To find the smallest number of eggs that Ethel could have had, we need to look for the common factor shared by the numbers 2, 3, 4, 5, and 6 that leaves a remainder of 1 when the eggs are taken out two at a time.
Let's start by finding the common factor of 2, 3, 4, 5, and 6. Here's a step-by-step process you can follow:
Step 1: Take the smallest pair of numbers, 2 and 3.
- Start with 2 eggs, pick them out two at a time. We have 1 egg remaining.
- Repeat this process with 3 eggs. We again have 1 egg remaining.
Step 2: Move to the next number, 4.
- Take 4 eggs and pick them out two at a time. We have 0 eggs remaining.
At this point, we know that the common factor of 2, 3, and 4 is 2. Since there is no remainder when Ethel takes the eggs out 2 at a time with 4 eggs in the basket, we need to check if this pattern continues with larger numbers.
Step 3: Repeat the process with 5.
- Take 5 eggs and pick them out two at a time. We have 1 egg remaining.
Step 4: Repeat the process with 6.
- Take 6 eggs and pick them out two at a time. We have 1 egg remaining.
Now we have found that the common factor for 2, 3, 4, 5, and 6 is 2, and there is always 1 egg left when picking them out 2, 3, 4, 5, or 6 at a time. To find the smallest number of eggs Ethel could have had, we need to find the least common multiple (LCM) of these numbers.
The LCM of 2, 3, 4, 5, and 6 is 60. This means that when Ethel had a basket of 60 eggs, she would have been left with 1 egg when picking them out 2, 3, 4, 5, or 6 at a time.
However, the question states that she picked them out 7 at a time and they came out even. This means that she must have had more than 60 eggs. To find the smallest number of eggs, we can keep adding 60 to find the next number that satisfies the condition.
Thus, the smallest number of eggs Ethel could have had is 60 + 60 = 120.