Some children were sharing oranges. If each child took 3 oranges, there would be 2 oranges left over. But if each child took 4 oranges, there would be 2 oranges short. How many oranges were there?
17 years ago
17 years ago
PLS HELP!!!
Some children were sharing oranges. If each child took 3 oranges, there would be 2 oranges left over. But if each child took 4 oranges, there would be 2 oranges short. How many oranges were there?
1 year ago
To solve this problem, let's represent the number of children as "c" and the total number of oranges as "o."
According to the problem, if each child took 3 oranges, there would be 2 oranges left over. This means that the number of oranges is 2 more than a multiple of 3. We can express this as:
o ≡ 2 (mod 3) -- Equation 1
Similarly, if each child took 4 oranges, there would be 2 oranges short. This means that the number of oranges is 2 less than a multiple of 4. We can express this as:
o ≡ -2 (mod 4) -- Equation 2
Now, we can use these two congruence equations to find the number of oranges. To do this, we need to find the least common multiple (LCM) of 3 and 4.
The LCM of 3 and 4 is 12.
By adding Equation 1 and Equation 2 together, we can find the number of oranges:
o ≡ 2 (mod 3)
o ≡ -2 (mod 4)
--------------------
o + o ≡ 2 + (-2) (mod 3) + (mod 4)
2o ≡ 0 (mod 12)
Simplifying the congruence equation:
2o ≡ 0 (mod 12)
Since the coefficient of "o" is 2, we need to find the inverse modulo 12:
2 * 6 ≡ 12 (mod 12)
Therefore, the inverse of 2 modulo 12 is 6.
Multiplying both sides of the congruence equation by the inverse of 2 modulo 12:
(2o * 6) ≡ (0 * 6) (mod 12)
12o ≡ 0 (mod 12)
Reducing the equation:
12o ≡ 0 (mod 12)
o ≡ 0 (mod 12)
This means that the number of oranges is a multiple of 12.
To find the least positive solution, we can substitute different values for "o" into each equation and look for a solution that satisfies both equations.
If we substitute o = 12 into the equations, we get:
Equation 1: 12 ≡ 2 (mod 3) -- This is true because 12 divided by 3 leaves a remainder of 0, which is equivalent to 2 (mod 3).
Equation 2: 12 ≡ -2 (mod 4) -- This is not true because 12 divided by 4 leaves a remainder of 0, not -2.
If we substitute o = 24 into the equations, we get:
Equation 1: 24 ≡ 2 (mod 3) -- This is not true because 24 divided by 3 leaves a remainder of 0, not 2.
Equation 2: 24 ≡ -2 (mod 4) -- This is not true because 24 divided by 4 leaves a remainder of 0, not -2.
If we substitute o = 36 into the equations, we get:
Equation 1: 36 ≡ 2 (mod 3) -- This is not true because 36 divided by 3 leaves a remainder of 0, not 2.
Equation 2: 36 ≡ -2 (mod 4) -- This is true because 36 divided by 4 leaves a remainder of 2, which is equivalent to -2 (mod 4).
Therefore, the least positive solution for "o" that satisfies both equations is o = 36.
Hence, there were 36 oranges.