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DIRECTIONS for the question:
Solve the following question and mark the best possible option.
After the division of a number successively by 3, 4, and 7, the remainders obtained are 2, 1 and 4 respectively. What will be the remainder if 84 divide the same number?
Solve the following question and mark the best possible option.
After the division of a number successively by 3, 4, and 7, the remainders obtained are 2, 1 and 4 respectively. What will be the remainder if 84 divide the same number?
- a)80
- b)76
- c)41
- d)53
Correct answer is option 'D'. Can you explain this answer?
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DIRECTIONS for the question:Solve the following question and mark the ...
►Since Dividend = Divisor � Quotient + Remainder, we have three divisors D1, D2, D3, as 3, 4, 7 and their respective remainders as R1, R2, R3 as 2, 1, 4.
►Taking quotient Q3 = 1, we have D3 � Q3 + R3 = 7 � 1 + 4 = 11. Now 11 becomes Q2 for D2 and so on.
►Now solving 4 � 11 + 1 = 45, 3 � 45 + 2 = 137.
►Now when 84 divides 137 remainder is 53.
►Now when 84 divides 137 remainder is 53.
Hence Option D is correct
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Most Upvoted Answer
DIRECTIONS for the question:Solve the following question and mark the ...
►Since Dividend = Divisor × Quotient + Remainder, we have three divisors D1, D2, D3, as 3, 4, 7 and their respective remainders as R1, R2, R3 as 2, 1, 4.
►Taking quotient Q3 = 1, we have D3 × Q3 + R3 = 7 × 1 + 4 = 11. Now 11 becomes Q2 for D2 and so on.
►Now solving 4 × 11 + 1 = 45, 3 × 45 + 2 = 137.
►Now when 84 divides 137 remainder is 53.
►Now when 84 divides 137 remainder is 53.
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DIRECTIONS for the question:Solve the following question and mark the ...
Understanding the Problem
When a number is divided successively by 3, 4, and 7, we obtain specific remainders. Let’s denote the number as 'N'.
Given Conditions:
- When N is divided by 3, the remainder is 2.
- When N is divided by 4, the remainder is 1.
- When N is divided by 7, the remainder is 4.
Mathematical Representation:
We can write these conditions in modular arithmetic:
- N ≡ 2 (mod 3)
- N ≡ 1 (mod 4)
- N ≡ 4 (mod 7)
Finding the General Solution
To solve these congruences, we can express N in terms of a single variable. Start with the largest modulo:
1. From N ≡ 4 (mod 7), we can express N as:
N = 7k + 4 for some integer k.
2. Substitute this into the second condition (N ≡ 1 (mod 4)):
- 7k + 4 ≡ 1 (mod 4)
- Simplifying gives: 3k ≡ 1 (mod 4).
- The multiplicative inverse of 3 mod 4 is 3. So, k ≡ 3 (mod 4).
- Therefore, k = 4m + 3 for some integer m.
3. Substitute back to find N:
N = 7(4m + 3) + 4 = 28m + 25.
4. Now, substitute into the first condition (N ≡ 2 (mod 3)):
- 28m + 25 ≡ 2 (mod 3).
- Since 28 ≡ 1 and 25 ≡ 1 (mod 3), we have:
- m + 1 ≡ 2 (mod 3) => m ≡ 1 (mod 3).
- Thus, m = 3n + 1 for some integer n.
5. Substitute back to find N:
N = 28(3n + 1) + 25 = 84n + 53.
Final Calculation
Now, we need to find the remainder when N is divided by 84:
- N = 84n + 53, hence the remainder is 53.
Conclusion
The remainder when this number is divided by 84 is 53. Thus, the correct answer is option D.
When a number is divided successively by 3, 4, and 7, we obtain specific remainders. Let’s denote the number as 'N'.
Given Conditions:
- When N is divided by 3, the remainder is 2.
- When N is divided by 4, the remainder is 1.
- When N is divided by 7, the remainder is 4.
Mathematical Representation:
We can write these conditions in modular arithmetic:
- N ≡ 2 (mod 3)
- N ≡ 1 (mod 4)
- N ≡ 4 (mod 7)
Finding the General Solution
To solve these congruences, we can express N in terms of a single variable. Start with the largest modulo:
1. From N ≡ 4 (mod 7), we can express N as:
N = 7k + 4 for some integer k.
2. Substitute this into the second condition (N ≡ 1 (mod 4)):
- 7k + 4 ≡ 1 (mod 4)
- Simplifying gives: 3k ≡ 1 (mod 4).
- The multiplicative inverse of 3 mod 4 is 3. So, k ≡ 3 (mod 4).
- Therefore, k = 4m + 3 for some integer m.
3. Substitute back to find N:
N = 7(4m + 3) + 4 = 28m + 25.
4. Now, substitute into the first condition (N ≡ 2 (mod 3)):
- 28m + 25 ≡ 2 (mod 3).
- Since 28 ≡ 1 and 25 ≡ 1 (mod 3), we have:
- m + 1 ≡ 2 (mod 3) => m ≡ 1 (mod 3).
- Thus, m = 3n + 1 for some integer n.
5. Substitute back to find N:
N = 28(3n + 1) + 25 = 84n + 53.
Final Calculation
Now, we need to find the remainder when N is divided by 84:
- N = 84n + 53, hence the remainder is 53.
Conclusion
The remainder when this number is divided by 84 is 53. Thus, the correct answer is option D.
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DIRECTIONS for the question:Solve the following question and mark the best possible option.After the division of a number successively by 3, 4, and 7, the remainders obtained are 2, 1 and 4 respectively. What will be the remainder if 84 divide the same number?a)80b)76c)41d)53Correct answer is option 'D'. Can you explain this answer?
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