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Two numbers when divided by a certain divisor leaves remainder of 14 and 19 respectively. Now when the sum of these two numbers is divided by the same divisor it leaves 12 as remainder. Find the divisor.
- a)33
- b)12
- c)21
- d)53
Correct answer is option 'C'. Can you explain this answer?
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Two numbers when divided by a certain divisor leaves remainder of 14 a...
Understanding the Problem
We have two numbers, let’s call them A and B. When divided by a divisor D, they leave remainders of 14 and 19, respectively. This can be expressed as:
- A = k1 * D + 14
- B = k2 * D + 19
where k1 and k2 are some integers.
When we sum these numbers (A + B) and divide by the same divisor D, it leaves a remainder of 12. This can be framed as:
- A + B = (k1 * D + 14) + (k2 * D + 19) = (k1 + k2) * D + 33
Now, we need to analyze the remainder:
Finding the Remainder
When A + B is divided by D:
- Remainder = (33) mod D
According to the problem, this remainder is equal to 12:
- (33) mod D = 12
This implies:
Setting Up the Equation
We can derive the equation:
- 33 - 12 = n * D (where n is some integer)
Simplifying gives us:
- 21 = n * D
This means D must be a divisor of 21.
Identifying Possible Divisors
The divisors of 21 are 1, 3, 7, and 21.
However, we need to ensure the remainders 14 and 19 are valid:
- D must be greater than both remainders (14 and 19) for the conditions to hold.
Conclusion
Thus, the only divisor that meets all conditions is:
- D = 21
Therefore, the correct answer is option 'C'.
We have two numbers, let’s call them A and B. When divided by a divisor D, they leave remainders of 14 and 19, respectively. This can be expressed as:
- A = k1 * D + 14
- B = k2 * D + 19
where k1 and k2 are some integers.
When we sum these numbers (A + B) and divide by the same divisor D, it leaves a remainder of 12. This can be framed as:
- A + B = (k1 * D + 14) + (k2 * D + 19) = (k1 + k2) * D + 33
Now, we need to analyze the remainder:
Finding the Remainder
When A + B is divided by D:
- Remainder = (33) mod D
According to the problem, this remainder is equal to 12:
- (33) mod D = 12
This implies:
Setting Up the Equation
We can derive the equation:
- 33 - 12 = n * D (where n is some integer)
Simplifying gives us:
- 21 = n * D
This means D must be a divisor of 21.
Identifying Possible Divisors
The divisors of 21 are 1, 3, 7, and 21.
However, we need to ensure the remainders 14 and 19 are valid:
- D must be greater than both remainders (14 and 19) for the conditions to hold.
Conclusion
Thus, the only divisor that meets all conditions is:
- D = 21
Therefore, the correct answer is option 'C'.
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