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In a division, the remainder was 71. With the same divisor, but twice the dividend, the remainder is 43. Which of the following numbers is the divisor?
- a)86
- b)93
- c)94
- d)99
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
In a division, the remainder was 71. With the same divisor, but twice ...
To solve this problem, we need to apply the concept of remainders and divisors. Let's break down the problem step by step:
Step 1: Let's assume the divisor as 'x' and the dividend as 'y'.
Step 2: According to the problem, when y is divided by x, the remainder is 71. This can be represented as:
y ≡ 71 (mod x)
Step 3: The problem also states that when twice the dividend (2y) is divided by the same divisor (x), the remainder is 43. This can be represented as:
2y ≡ 43 (mod x)
Step 4: Now, we need to solve these two congruence equations to find the value of the divisor 'x'.
Solving the first equation:
y ≡ 71 (mod x)
This implies y = ax + 71 (where a is an integer)
Solving the second equation:
2y ≡ 43 (mod x)
This implies 2y = bx + 43 (where b is an integer)
Step 5: We can substitute the value of y from the first equation into the second equation:
2(ax + 71) = bx + 43
2ax + 142 = bx + 43
2ax - bx = 43 - 142
(a - b)x = -99
Step 6: Since a and b are integers, for the equation (a - b)x = -99 to hold true, x must be a factor of -99.
Step 7: Checking the options given:
a) 86 is not a factor of -99
b) 93 is not a factor of -99
c) 94 is not a factor of -99
d) 99 is a factor of -99
Hence, the correct answer is option 'D' - 99.
In conclusion, by applying the concept of remainders and divisors, we determined that the divisor in the given problem is 99.
Step 1: Let's assume the divisor as 'x' and the dividend as 'y'.
Step 2: According to the problem, when y is divided by x, the remainder is 71. This can be represented as:
y ≡ 71 (mod x)
Step 3: The problem also states that when twice the dividend (2y) is divided by the same divisor (x), the remainder is 43. This can be represented as:
2y ≡ 43 (mod x)
Step 4: Now, we need to solve these two congruence equations to find the value of the divisor 'x'.
Solving the first equation:
y ≡ 71 (mod x)
This implies y = ax + 71 (where a is an integer)
Solving the second equation:
2y ≡ 43 (mod x)
This implies 2y = bx + 43 (where b is an integer)
Step 5: We can substitute the value of y from the first equation into the second equation:
2(ax + 71) = bx + 43
2ax + 142 = bx + 43
2ax - bx = 43 - 142
(a - b)x = -99
Step 6: Since a and b are integers, for the equation (a - b)x = -99 to hold true, x must be a factor of -99.
Step 7: Checking the options given:
a) 86 is not a factor of -99
b) 93 is not a factor of -99
c) 94 is not a factor of -99
d) 99 is a factor of -99
Hence, the correct answer is option 'D' - 99.
In conclusion, by applying the concept of remainders and divisors, we determined that the divisor in the given problem is 99.
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Community Answer
In a division, the remainder was 71. With the same divisor, but twice ...
Let the dividend be x and the remainder be 71, so that x + 71 is divisible by the divisor.
When it is done, it comes twice, i.e. 2x + 142, which is also divisible.
According to the question, it leaves the remainder 43.
So, 2x + 142 – 43 = 2x + 99
So, the divisor is 99.
When it is done, it comes twice, i.e. 2x + 142, which is also divisible.
According to the question, it leaves the remainder 43.
So, 2x + 142 – 43 = 2x + 99
So, the divisor is 99.
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In a division, the remainder was 71. With the same divisor, but twice the dividend, the remainder is 43. Which of the following numbers is the divisor?a)86b)93c)94d)99Correct answer is option 'D'. Can you explain this answer?
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