Aliaba was arranging the carpets for the flying carpet brigade They were going to fly in a rectangular formation when he put 4 carpets in each row he ad 3 left over . When he put 5 carpets in each row he had one left over. How many carpets could there have been in the carpet brigade ?
divide by five, remainder 1
6, 11, 16, 21, 26, 31 , .......
divide by 4, remainder 3
7 , 11 , 15 , 19 , 23 , 27 , 31 score !
31 makes 7 rows of 4 with 3 left over
31 makes 6 rows of 5 with 1 let over
11 is just the first one :) There are lots.
There are 6 carpets
To find out how many carpets could there have been in the carpet brigade, we can use a system of equations.
Let's say the total number of carpets in the brigade is represented by "x".
According to the problem, when Aliaba put 4 carpets in each row, he had 3 left over. This means that x must be congruent to 3 (mod 4), which can be represented as:
x ≡ 3 (mod 4)
Similarly, when Aliaba put 5 carpets in each row, he had 1 left over. This gives us another congruence:
x ≡ 1 (mod 5)
Now, we can solve this system of congruences to find the possible values of x.
Let's start by finding the first few positive integers that satisfy the first congruence (x ≡ 3 (mod 4)):
x = 3, 7, 11, 15, 19, 23, ...
Next, let's check which of these values satisfy the second congruence (x ≡ 1 (mod 5)):
Only 11 and 23 satisfy the second congruence.
Therefore, there could have been either 11 or 23 carpets in the carpet brigade.