Math Problem Statement
MBA Nexus - CAT & OMETS 2024, [19-10-2024 08:49] Q3 A 4-digit number when successively divided by 9, 8 & 6 leaves remainder of 4, 3, & 2 respectively. What could be the remainder when the number is divided by 20?
MBA Nexus - CAT & OMETS 2024, [19-10-2024 08:50] [ Poll : Ans ]
- 19
- 11
- 15
- All of the above
Solution
We are tasked with finding a 4-digit number that, when successively divided by 9, 8, and 6, leaves remainders 4, 3, and 2, respectively. Ultimately, we want to determine the remainder when this number is divided by 20.
Step-by-step solution:
Let's denote the number by . We are given the following conditions:
We will solve this system using the Chinese Remainder Theorem.
Step 1: Solve the first two congruences
We have:
Let's assume for some integer (from ). Substituting this into the second congruence:
Simplifying: Since , we have: Thus, for some integer .
Step 2: Substitute back into the equation for
Now substitute into : Thus:
Step 3: Solve the third congruence
We now incorporate the third condition: Substitute into this congruence: Since and , we have: This is clearly false, so let's correct the third modulus condition. Instead, we handle the modulus 6 step by looking for compatible simultaneous conditions.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Number Theory
Modular Arithmetic
Chinese Remainder Theorem
Formulas
N ≡ r (mod m)
Chinese Remainder Theorem conditions
Theorems
Chinese Remainder Theorem
Suitable Grade Level
College/University Level
Related Recommendation
Solving Modular Arithmetic Problem with Chinese Remainder Theorem
Solve 3-Digit Numbers with Specific Remainders Using Modular Arithmetic
Solve a 4-Digit Number Problem Using the Chinese Remainder Theorem
Find the Minimum 3-Digit Number Divisible by 7, 5, and 4 with a Remainder of 2
Number Theory Problem: Dividing 390390 by 77 and 1111