The number of apples in a basket is between 60 and 120. If i put these apples into bags of 5, there will be a remainder of 1 apple. If i put these apples into bags of 8, there will be a remainder of 6 apples. Find the total number of apples in the basket.
I managed to find out the answer using Guess and check method. Please kindly give the solution in other easier methods.
Thank you.
To solve this problem, we can use a system of equations. Let's denote the number of apples as "a".
First, let's consider the remainder after dividing "a" by 5. We know that when divided by 5, there is a remainder of 1. So we can express this as:
a ≡ 1 (mod 5)
Secondly, let's consider the remainder after dividing "a" by 8. We know that when divided by 8, there is a remainder of 6. So we can express this as:
a ≡ 6 (mod 8)
To find the total number of apples in the basket, we need to find a value of "a" that satisfies both of these equations.
To make this process easier, we can simplify the equations by subtracting the remainder from both sides:
a - 1 ≡ 0 (mod 5)
a - 6 ≡ 0 (mod 8)
Now, we can express these simplified equations as a system of equations:
a ≡ 1 (mod 5)
a ≡ 6 (mod 8)
There are different methods to solve this system, one of which is the Chinese Remainder Theorem. However, for simplicity, we can use a trial and error approach combined with the concept of the least common multiple (LCM).
First, let's find the LCM of 5 and 8. The LCM of two numbers is the smallest multiple that is divisible by both numbers. In this case, the LCM of 5 and 8 is 40.
Now, let's find a number that satisfies both equations. We can start by checking multiples of the LCM (40) and adding 1 to each multiple until we find a satisfying value:
40 × 0 + 1 = 1 (not satisfying both equations)
40 × 1 + 1 = 41 (not satisfying both equations)
40 × 2 + 1 = 81 (satisfying both equations)
So, the total number of apples in the basket is 81.
In conclusion, the total number of apples in the basket is 81, obtained by solving the system of equations using the concept of the least common multiple (LCM) and trial and error method.