Mr.Baker has baked some muffins. If he packs them in boxes of 4, he will have 3 left over. If he packs them in boxes of 5, he will also have 3 left over If he packs them in boxes of 6, he will have only 1 left over. Find the least possible number of muffins Mr.Baker has baked.
numbers which when divided by 4 leave 3:
7 11 15 19 23 27 31 35 39 43 ...
numbers which when divided by 5 leave 3:
8 13 18 23 28 33 38 43 48 ...
numbers which when divided by 6 leave 1:
7 13 19 25 31 37 43 ...
ahh, 43 is in all 3 lists, so ....
yes
They aren’t giving the answer
Yes they are, it's 43.
To find the least possible number of muffins Mr. Baker has baked, we need to find a number that satisfies all three conditions. Let's denote the number of muffins as "x".
According to the first condition, if Mr. Baker packs the muffins in boxes of 4, he will have 3 left over. This can be written as the equation: x ≡ 3 (mod 4).
Similarly, from the second condition, we have the equation: x ≡ 3 (mod 5).
And from the third condition, the equation is: x ≡ 1 (mod 6).
To find the least possible value of x that satisfies all three equations, we can use the Chinese Remainder Theorem (CRT). CRT is a mathematical method that allows us to solve systems of linear congruences.
First, let's reduce each equation to the smallest possible positive integer solution:
For the first equation, the possible solutions are 3, 7, 11, 15, ...
For the second equation, the possible solutions are 3, 8, 13, 18, ...
For the third equation, the possible solutions are 1, 7, 13, 19, ...
Now, using CRT, we need to find the smallest solution that satisfies all three equations. To do this, we need to find a number that is congruent to 3 (mod 4), 3 (mod 5), and 1 (mod 6) simultaneously.
By calculating the least common multiple (LCM) of 4, 5, and 6, we find that LCM(4, 5, 6) = 60.
Now, let's find a number that satisfies all three equations:
Starting from 3 (which satisfies all equations individually), we can add 60 to it repeatedly until we find the smallest number that satisfies all equations.
By adding 60 to 3, we get 63, which satisfies all equations: 63 ≡ 3 (mod 4), 63 ≡ 3 (mod 5), and 63 ≡ 1 (mod 6).
Therefore, the least possible number of muffins Mr. Baker has baked is 63.