Math Problem Statement
How many integers between 1 and 100 leave a remainder of 3 when divided by 5 and a remainder of 5 when divided by 7?
Solution
We are tasked with finding how many integers between 1 and 100 leave:
- a remainder of 3 when divided by 5, and
- a remainder of 5 when divided by 7.
Let’s break this down step by step.
Step 1: Represent the conditions as congruences
The conditions can be written as two congruences:
- (i.e., leaves a remainder of 3 when divided by 5)
- (i.e., leaves a remainder of 5 when divided by 7)
Step 2: Solve the system of congruences using the Chinese Remainder Theorem
The system of congruences is:
We will solve this system step by step.
Step 3: Use the first congruence
Let for some integer . We now substitute this into the second congruence.
Step 4: Substitute into the second congruence
Substitute into : Simplifying:
Step 5: Solve
Now, we need to solve . To do this, we can multiply both sides by the modular inverse of 5 modulo 7.
The inverse of 5 modulo 7 can be found using trial and error or extended Euclidean algorithm. The inverse of 5 modulo 7 is 3, because:
Now multiply both sides of the equation by 3:
Thus, for some integer .
Step 6: Substitute back into
Substitute into :
Step 7: Find values of between 1 and 100
We need to find the integers that are between 1 and 100. Solving for : Subtract 33 from all parts: Divide by 35:
Step 8: Calculate the corresponding values
For : For :
Step 9: Conclusion
The integers between 1 and 100 that satisfy both conditions are and . Therefore, there are 2 integers that satisfy the given conditions.
Do you need more details or have any questions? Here are some related problems you might find interesting:
- How many integers between 1 and 200 leave a remainder of 4 when divided by 6 and 3 when divided by 8?
- What are the smallest three integers that satisfy and ?
- How many integers between 1 and 500 satisfy and ?
- Solve the system and .
- How many integers between 1 and 1000 satisfy and ?
Tip: The Chinese Remainder Theorem is a powerful tool for solving systems of simultaneous congruences.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Modular Arithmetic
Chinese Remainder Theorem
Formulas
x ≡ 3 (mod 5)
x ≡ 5 (mod 7)
Theorems
Chinese Remainder Theorem
Suitable Grade Level
Grades 10-12
Related Recommendation
Solve a Modular Arithmetic Problem Using Chinese Remainder Theorem
Solving Congruence Systems: Finding a Number That Satisfies Multiple Remainder Conditions
Finding a Number Using the Chinese Remainder Theorem: Divided by 3, 5, and 8
Solve 3-Digit Numbers with Specific Remainders Using Modular Arithmetic
Solving Modular Arithmetic Problem with Chinese Remainder Theorem