Discover a wealth of knowledge at Westonci.ca, where experts provide answers to your most pressing questions. Discover comprehensive solutions to your questions from a wide network of experts on our user-friendly platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
To solve the problem, we need to find a number of eggs that satisfies the following conditions:
1. When the eggs are removed in groups of 3, there is 1 egg left.
2. When the eggs are removed in groups of 5, there is 1 egg left.
3. When the eggs are removed in groups of 7, there are no eggs left.
4. The total number of eggs is less than or equal to 100.
These conditions can be translated into a set of congruences:
[tex]\[ \text{(i)} \quad N \equiv 1 \pmod{3} \][/tex]
[tex]\[ \text{(ii)} \quad N \equiv 1 \pmod{5} \][/tex]
[tex]\[ \text{(iii)} \quad N \equiv 0 \pmod{7} \][/tex]
### Step-by-step Solution:
1. Let's consider the third condition first; this tells us that the number of eggs [tex]\( N \)[/tex] must be a multiple of 7. Therefore, we can write:
[tex]\[ N = 7k \text{ for some integer } k \][/tex]
2. Next, using the first condition, [tex]\( N \equiv 1 \pmod{3} \)[/tex], we substitute [tex]\( N \)[/tex] and get:
[tex]\[ 7k \equiv 1 \pmod{3} \][/tex]
This simplifies to:
[tex]\[ 7k \mod 3 = 1 \][/tex]
[tex]\[ k \equiv 1 \pmod{3} \][/tex]
[tex]\[ \Rightarrow k = 3m + 1 \text{ for some integer } m \][/tex]
Therefore, substituting back for [tex]\( N \)[/tex]:
[tex]\[ N = 7(3m + 1) \][/tex]
[tex]\[ N = 21m + 7 \][/tex]
3. Now we use the second condition, [tex]\( N \equiv 1 \pmod{5} \)[/tex], to find [tex]\( m \)[/tex]. Substitute [tex]\( N \)[/tex]:
[tex]\[ 21m + 7 \equiv 1 \pmod{5} \][/tex]
Simplifying, we have:
[tex]\[ 21m + 7 \equiv 1 \pmod{5} \][/tex]
[tex]\[ 21m \equiv -6 \pmod{5} \][/tex]
[tex]\[ 21m \equiv -1 \pmod{5} \][/tex]
[tex]\[ 21 \equiv 1 \pmod{5} \][/tex]
Hence:
[tex]\[ m \equiv -1 \pmod{5} \][/tex]
[tex]\[ m = 5n - 1 \text{ for some integer } n \][/tex]
4. Finally, substituting [tex]\( m \)[/tex] back into the equation for [tex]\( N \)[/tex]:
[tex]\[ N = 21(5n - 1) + 7 \][/tex]
[tex]\[ N = 105n - 21 + 7 \][/tex]
[tex]\[ N = 105n - 14 \][/tex]
Since the number of eggs must be less than or equal to 100:
[tex]\[ 105n - 14 \leq 100 \][/tex]
[tex]\[ 105n \leq 114 \][/tex]
[tex]\[ n \leq \frac{114}{105} \][/tex]
[tex]\[ n \leq 1.0857 \][/tex]
Since [tex]\( n \)[/tex] must be an integer, [tex]\( n = 1 \)[/tex]:
Hence,
[tex]\[ N = 105(1) - 14 \][/tex]
[tex]\[ N = 91 \][/tex]
Therefore, the woman has [tex]\(\boxed{91}\)[/tex] eggs.
1. When the eggs are removed in groups of 3, there is 1 egg left.
2. When the eggs are removed in groups of 5, there is 1 egg left.
3. When the eggs are removed in groups of 7, there are no eggs left.
4. The total number of eggs is less than or equal to 100.
These conditions can be translated into a set of congruences:
[tex]\[ \text{(i)} \quad N \equiv 1 \pmod{3} \][/tex]
[tex]\[ \text{(ii)} \quad N \equiv 1 \pmod{5} \][/tex]
[tex]\[ \text{(iii)} \quad N \equiv 0 \pmod{7} \][/tex]
### Step-by-step Solution:
1. Let's consider the third condition first; this tells us that the number of eggs [tex]\( N \)[/tex] must be a multiple of 7. Therefore, we can write:
[tex]\[ N = 7k \text{ for some integer } k \][/tex]
2. Next, using the first condition, [tex]\( N \equiv 1 \pmod{3} \)[/tex], we substitute [tex]\( N \)[/tex] and get:
[tex]\[ 7k \equiv 1 \pmod{3} \][/tex]
This simplifies to:
[tex]\[ 7k \mod 3 = 1 \][/tex]
[tex]\[ k \equiv 1 \pmod{3} \][/tex]
[tex]\[ \Rightarrow k = 3m + 1 \text{ for some integer } m \][/tex]
Therefore, substituting back for [tex]\( N \)[/tex]:
[tex]\[ N = 7(3m + 1) \][/tex]
[tex]\[ N = 21m + 7 \][/tex]
3. Now we use the second condition, [tex]\( N \equiv 1 \pmod{5} \)[/tex], to find [tex]\( m \)[/tex]. Substitute [tex]\( N \)[/tex]:
[tex]\[ 21m + 7 \equiv 1 \pmod{5} \][/tex]
Simplifying, we have:
[tex]\[ 21m + 7 \equiv 1 \pmod{5} \][/tex]
[tex]\[ 21m \equiv -6 \pmod{5} \][/tex]
[tex]\[ 21m \equiv -1 \pmod{5} \][/tex]
[tex]\[ 21 \equiv 1 \pmod{5} \][/tex]
Hence:
[tex]\[ m \equiv -1 \pmod{5} \][/tex]
[tex]\[ m = 5n - 1 \text{ for some integer } n \][/tex]
4. Finally, substituting [tex]\( m \)[/tex] back into the equation for [tex]\( N \)[/tex]:
[tex]\[ N = 21(5n - 1) + 7 \][/tex]
[tex]\[ N = 105n - 21 + 7 \][/tex]
[tex]\[ N = 105n - 14 \][/tex]
Since the number of eggs must be less than or equal to 100:
[tex]\[ 105n - 14 \leq 100 \][/tex]
[tex]\[ 105n \leq 114 \][/tex]
[tex]\[ n \leq \frac{114}{105} \][/tex]
[tex]\[ n \leq 1.0857 \][/tex]
Since [tex]\( n \)[/tex] must be an integer, [tex]\( n = 1 \)[/tex]:
Hence,
[tex]\[ N = 105(1) - 14 \][/tex]
[tex]\[ N = 91 \][/tex]
Therefore, the woman has [tex]\(\boxed{91}\)[/tex] eggs.
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
