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Two numbers when divided by the same divisor leave remainders 422 and 246. When sum of these two numbers is divided by the same divisor, the remainder is 112. What is the divisor?
- a)426
- b)556
- c)526
- d)400
Correct answer is option 'B'. Can you explain this answer?
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Two numbers when divided by the same divisor leave remainders 422 and...
Problem:
Two numbers, when divided by the same divisor, leave remainders of 422 and 246. When the sum of these two numbers is divided by the same divisor, the remainder is 112. What is the divisor?
Solution:
To solve this problem, we need to use the concept of the remainder theorem. Let's break down the problem into steps:
Step 1: Let's assume the two numbers to be x and y.
The first number, x, when divided by the divisor, leaves a remainder of 422.
The second number, y, when divided by the same divisor, leaves a remainder of 246.
Step 2: The sum of these two numbers, x + y, when divided by the same divisor, leaves a remainder of 112.
Step 3: Now, let's express the given information in the form of equations.
Equation 1: x ≡ 422 (mod d), where d is the divisor.
Equation 2: y ≡ 246 (mod d), where d is the divisor.
Equation 3: (x + y) ≡ 112 (mod d), where d is the divisor.
Step 4: We can subtract Equation 2 from Equation 1 to eliminate the variable y.
(x - y) ≡ (422 - 246) (mod d)
x - y ≡ 176 (mod d) ---(Equation 4)
Step 5: Now, let's add Equation 4 to Equation 3 to eliminate the variable x.
(x - y) + (x + y) ≡ 176 + 112 (mod d)
2x ≡ 288 (mod d)
x ≡ 144 (mod d) ---(Equation 5)
Step 6: We can substitute Equation 5 into Equation 1 to solve for d.
144 ≡ 422 (mod d)
278 ≡ 0 (mod d)
Step 7: Now, we need to find the divisor, d, such that 278 ≡ 0 (mod d).
To find the divisor, we need to find the factors of 278 and check which one satisfies the equation.
Factors of 278: 1, 2, 139, 278
Among the factors, only 139 satisfies the equation 278 ≡ 0 (mod d).
Therefore, the divisor is 139.
Conclusion:
The divisor is 139, which is not listed as an option. However, the correct answer must be one of the given options. Therefore, we made an error in our calculations or the given options are incorrect.
Two numbers, when divided by the same divisor, leave remainders of 422 and 246. When the sum of these two numbers is divided by the same divisor, the remainder is 112. What is the divisor?
Solution:
To solve this problem, we need to use the concept of the remainder theorem. Let's break down the problem into steps:
Step 1: Let's assume the two numbers to be x and y.
The first number, x, when divided by the divisor, leaves a remainder of 422.
The second number, y, when divided by the same divisor, leaves a remainder of 246.
Step 2: The sum of these two numbers, x + y, when divided by the same divisor, leaves a remainder of 112.
Step 3: Now, let's express the given information in the form of equations.
Equation 1: x ≡ 422 (mod d), where d is the divisor.
Equation 2: y ≡ 246 (mod d), where d is the divisor.
Equation 3: (x + y) ≡ 112 (mod d), where d is the divisor.
Step 4: We can subtract Equation 2 from Equation 1 to eliminate the variable y.
(x - y) ≡ (422 - 246) (mod d)
x - y ≡ 176 (mod d) ---(Equation 4)
Step 5: Now, let's add Equation 4 to Equation 3 to eliminate the variable x.
(x - y) + (x + y) ≡ 176 + 112 (mod d)
2x ≡ 288 (mod d)
x ≡ 144 (mod d) ---(Equation 5)
Step 6: We can substitute Equation 5 into Equation 1 to solve for d.
144 ≡ 422 (mod d)
278 ≡ 0 (mod d)
Step 7: Now, we need to find the divisor, d, such that 278 ≡ 0 (mod d).
To find the divisor, we need to find the factors of 278 and check which one satisfies the equation.
Factors of 278: 1, 2, 139, 278
Among the factors, only 139 satisfies the equation 278 ≡ 0 (mod d).
Therefore, the divisor is 139.
Conclusion:
The divisor is 139, which is not listed as an option. However, the correct answer must be one of the given options. Therefore, we made an error in our calculations or the given options are incorrect.
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Community Answer
Two numbers when divided by the same divisor leave remainders 422 and...
Suppose numbers are P & Q and divisor is A
So, we can say that
P= mA + 422
Q = nA + 246, where m & n are natural numbers.
If sum of P & Q is divided by A, then remainder will come from (422 + 246)/A
So we can say, 668 = 1*A+ 112
A=668- 112
A = 556. Hence, answer option is 2.
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Two numbers when divided by the same divisor leave remainders 422 and 246. When sum of these two numbers is divided by the same divisor, the remainder is 112. What is the divisor?a)426b)556c)526d)400Correct answer is option 'B'. Can you explain this answer?
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