A peddler is taking eggs to market to sell. The eggs are in a cart that holds up to 500 eggs. If the eggs are removed from the cart 2, 3, 4, 5,or 6 at a time, one egg is always left over. If the eggs are removed 7 at a time, no eggs are left over. How many eggs are in the cart.

Question

A peddler is taking eggs to market to sell. The eggs are in a cart that holds up to 500 eggs. If the eggs are removed from the cart 2, 3, 4, 5,or 6 at a time, one egg is always left over. If the eggs are removed 7 at a time, no eggs are left over. How many eggs are in the cart.

Answer 1

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Answer 2

To solve this problem, we need to find the total number of eggs in the cart. Let's break down the information given:

1. When the eggs are removed 2, 3, 4, 5, or 6 at a time, one egg is always left over. This means that the total number of eggs in the cart is one more than a multiple of 2, 3, 4, 5, or 6.
Let's consider the possible numbers of eggs in the cart:
- If the total number of eggs is a multiple of 2, the remainder will be 1.
- If the total number of eggs is a multiple of 3, the remainder will be 1.
- If the total number of eggs is a multiple of 4, the remainder will be 1.
- If the total number of eggs is a multiple of 5, the remainder will be 1.
- If the total number of eggs is a multiple of 6, the remainder will be 1.

2. When the eggs are removed 7 at a time, no eggs are left over. This means that the total number of eggs in the cart is a multiple of 7.

Now, let's find the solution. We need to look for a number that satisfies both conditions. We can start by finding a number that is a multiple of 7 and has a remainder of 1 when divided by 2, 3, 4, 5, and 6.

Consider the multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, ...

Now let's add 1 to each of these numbers and check if they are divisible by 2, 3, 4, 5, and 6.

For example:
- 7 + 1 = 8 (not divisible by 2 or 4)
- 14 + 1 = 15 (not divisible by 2 or 4)
- 21 + 1 = 22 (not divisible by 4)
- 28 + 1 = 29 (not divisible by 2, 4, or 6)

You can continue this process until you find a number that satisfies all the conditions.

After checking, we find that the first number that satisfies both conditions is 119.

Therefore, there are 119 eggs in the cart.