There are fewer than 33 eggs in a basket. If they are counted out 6 at a time, there
are 4 left over. If they are counted out 5 at a time, there are 3 left. How many eggs are in the basket?
not just an answer
6n+4: 10,16,22,28
5n+3: 8,13,18,23,28
we have a winner! 28
6x+4:10,16,22,28,34>33
5x+3:8,13,18,23,28,33=33
the no. of eggs is 28
To find out the number of eggs in the basket, we can use a system of equations. Let's start by assigning a variable to represent the unknown number of eggs in the basket.
Let's call the unknown number of eggs "x".
From the given information, we know that when the eggs are counted out 6 at a time, there are 4 left over. This can be represented by the equation:
x ≡ 4 (mod 6)
The symbol "≡" means "congruent to" and "mod" represents the modulus operation, which gives the remainder when dividing a number by another. In this case, the remainder when dividing x by 6 is 4.
Similarly, when the eggs are counted out 5 at a time, there are 3 left over, which can be represented by the equation:
x ≡ 3 (mod 5)
Now we have a system of equations:
x ≡ 4 (mod 6)
x ≡ 3 (mod 5)
To solve this system, we can use the Chinese Remainder Theorem or a method called "divisors of differences." In this case, we can use the divisors of differences method.
The difference between 6 and 5 is 1. The divisors of 1 are 1 and -1. We can use these divisors to find a solution.
First, let's find a solution for the equation x ≡ 4 (mod 6). We can start by adding multiples of 6 to 4 until we find a number that satisfies the equation:
4, 10, 16, 22, 28, 34
When we reach 34, we have a number that is congruent to 4 (mod 6). Now let's check if it also satisfies the equation x ≡ 3 (mod 5):
34 ≡ 4 (mod 6) and 34 ≡ 4 (mod 5)
Since 34 satisfies both equations, we have found a solution. Therefore, there are 34 eggs in the basket.
So, the answer is that there are 34 eggs in the basket.