Counting Chicken Wings
At Annie's Home-Cooked Chicken Wings Restaurant, chicken wings are served by the bucket. The Biggest Bucket O'Wings is really big! Figure out how many wings are in the bucket.
If the wings are removed two at a time, one wing will be left. If they're removed three at a time, two wings will remain. If they're removed four at a time, then three will remain; if they're removed five at a time, then four will remain; and if six at a time are taken from the bucket, then five wings will remain. If they're taken out seven at a time, no wings will be left over.
What's the smallest possible number of wings that could be in the bucket? EXPLAIN YOUR ANSWER THOROUGHLY!!!
The number must be evenly divisible by 7. It must also be odd, since taking two at a time leaves 1. It must be two more than a multiple of 3, etc..
What can work?
Not 7 (3 at a time leaves 1)
Not 21 (3 at a time leaves none)
35? No. Five at a time leaves none.
49? No. Three at a times leaves 1
63? No. Three a time leaves none
77? No. Four at a time leaves 1
91? No. Three at a time leaves 1
119? Try it.
LCM of 2,3,4,5,6=60《LCM 1》
Since the number of wings divided by 2,3,4,5,6 need one more to divide evenly,the number o wings must be
《LCM 1》×? -1
TRY ?=1
59
59÷7 IS NOT EVENLY DIVIDED
TRY ?=2
119
119÷7=17
so s bucket of chicken wings contains 119 chicken wings
It is quite big,isn't it?
the coach is building shelves to store and organize the team's collection of 175 game videos. how many shelves will be needed if one shelf holds 28 videos?
119
Why is it 119
119
To determine the smallest possible number of wings in the bucket, we need to find the least common multiple (LCM) of the numbers 2, 3, 4, 5, and 6, which represents the number of wings that can satisfy all the given conditions.
First, let's analyze the given conditions one by one:
1. If the wings are removed two at a time, one wing will be left.
This means the number of wings is 1 more than a multiple of 2. Since the wings cannot be negative, we start from 1.
2. If they're removed three at a time, two wings will remain.
This means the number of wings is 2 more than a multiple of 3.
3. If they're removed four at a time, three will remain.
This means the number of wings is 3 more than a multiple of 4.
4. If they're removed five at a time, four will remain.
This means the number of wings is 4 more than a multiple of 5.
5. If six at a time are taken from the bucket, then five wings will remain.
This means the number of wings is 5 more than a multiple of 6.
6. If they're taken out seven at a time, no wings will be left over.
This means the number of wings is a multiple of 7.
Now, we need to find the LCM of these conditions. To do this, we can start by listing the multiples of each number and finding their common multiples:
Multiples of 2: 2, 4, 6, 8, 10, 12, 14, ...
Multiples of 3: 3, 6, 9, 12, 15, 18, ...
Multiples of 4: 4, 8, 12, 16, 20, ...
Multiples of 5: 5, 10, 15, 20, ...
Multiples of 6: 6, 12, 18, 24, ...
Multiples of 7: 7, 14, 21, 28, ...
From these lists, we can see that the common multiples are 12 and 24. However, since we need to find the smallest possible number, we choose the smallest common multiple, which is 12.
Therefore, the smallest possible number of wings in the bucket is 12.
To recap, we found the least common multiple (LCM) of the numbers 2, 3, 4, 5, 6, and 7, which gave us the smallest value that satisfies all the given conditions. In this case, the LCM is 12, which means there are 12 wings in the bucket at Annie's Home-Cooked Chicken Wings Restaurant.