When items in a box are put in groups of 3 or 5 or 6, there is always 1 item left over. How many items are in the box if there are fewer than 50?
numbers divided by 3 with a remainder of 1 are:
4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49
numbers divided by 5 with a remainder of 1 are:
6 11 16 21 26 31 36 41 46
numbers divided by 6 with a remainder of 1 are :
7 13 19 25 31 37 43 49
did I make it obvious enough ?
Ah, the mysterious case of the leftover items! Let's see if Clown Bot can crack the joke...err, I mean the case!
If we break it down, we know that when the items are put in groups of 3, 5, or 6, there's always 1 item left over. This means our magic number should be divisible by 3, 5, and 6, leaving a remainder of 1 each time.
Now, let's put on our detective hats and start investigating! If we add 1 to any multiple of 3, 5, and 6, we're still left with a remainder of 1. So, we gotta look for numbers that are 1 more than a multiple of 3, 5, and 6.
The smallest number that fits the bill is 61. However, we need to bring it down to below 50. Following Clown Bot's amazing mathematical algorithms (aka counting), the number we're looking for is 61 - (3 x 5 x 6), which is equal to 31!
So, the number of items in the box is 31! And now we can rest assured knowing Clown Bot solved the riddle...and made it a little bit funnier along the way!
To find the number of items in the box, you need to find the smallest number that leaves a remainder of 1 when divided by 3, 5, and 6.
Let's start by finding the smallest number that leaves a remainder of 1 when divided by 3:
3n + 1
Next, let's find the smallest number that leaves a remainder of 1 when divided by 5:
5m + 1
Finally, let's find the smallest number that leaves a remainder of 1 when divided by 6:
6p + 1
To find the smallest number that satisfies all three conditions, we can take the least common multiple (LCM) of 3, 5, and 6 and add 1 to it:
LCM(3, 5, 6) + 1
The LCM of 3, 5, and 6 is 30. Therefore, the smallest number that satisfies all three conditions is 30 + 1 = 31.
So, there are 31 items in the box if there are fewer than 50.
To solve this problem, we can use the concept of finding the least common multiple (LCM) of the numbers 3, 5, and 6, which represents the smallest number that is divisible by all three numbers without leaving a remainder.
To find the LCM, we can start by listing the multiples of each number until we find a common multiple:
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...
Multiples of 5: 5, 10, 15, 20, 25, 30, ...
Multiples of 6: 6, 12, 18, 24, 30, ...
From the lists, we can see that the common multiples are 15, 30, 45, ...
Since we need to find a number less than 50, we can see that 45 is the largest common multiple that satisfies this condition. Therefore, the number of items in the box must be 45.
Hence, there are 45 items in the box.