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A natural number when divided by D leaves remainder 5. When four times of the number is divided with D, we get remainder as 8. Which of the following can be the value of D? (Given: D is a natural number.)
- a)12
- b)8
- c)6
- d)Both (1) and (2)
- e)Both (2) and (3)
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
A natural number when divided by D leaves remainder 5. When four times...
Let the number be N.
N = Dk + 5; where k is the quotient. 4N = 4Dk + 20 ...(i)
N = Dk + 5; where k is the quotient. 4N = 4Dk + 20 ...(i)
As we get 8 as the remainder when 4N is divided by D,
4N = Dy+ 8 ...(ii)
From (i) and (ii);
D(y - 4k) =12
L.H.S. is divisible by D ⇒R.H.S. is divisible by D.
⇒ D = 1 or 2 or 3 or 4 or 6 or 12
D should also be greater than 5 and 8.
From the given option only 12 satisfies the condition.
Hence, option 1.
From the given option only 12 satisfies the condition.
Hence, option 1.
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A natural number when divided by D leaves remainder 5. When four times...
Given information:
- A natural number when divided by D leaves a remainder of 5.
- When four times the number is divided by D, the remainder is 8.
To find: Possible values for D.
Solution:
Let's assume the natural number to be x.
First condition:
When x is divided by D, the remainder is 5.
This can be represented as x ≡ 5 (mod D), where '≡' represents congruence.
Second condition:
When four times x is divided by D, the remainder is 8.
This can be represented as 4x ≡ 8 (mod D).
Using the Chinese Remainder Theorem:
Since x ≡ 5 (mod D), we can express x as x = 5 + kD, where k is a positive integer.
Substituting this value of x in the second condition, we get:
4(5 + kD) ≡ 8 (mod D)
20 + 4kD ≡ 8 (mod D)
20 ≡ 8 (mod D)
Analysis of the second condition:
The equation 20 ≡ 8 (mod D) implies that D divides (20 - 8) = 12.
So, the possible values of D are factors of 12.
Possible values for D:
The factors of 12 are 1, 2, 3, 4, 6, and 12.
Out of these, only D = 12 satisfies the first condition x ≡ 5 (mod D) because any number divisible by 12 will have a remainder of 0 when divided by 12.
Therefore, the only possible value for D is 12 (option A).
- A natural number when divided by D leaves a remainder of 5.
- When four times the number is divided by D, the remainder is 8.
To find: Possible values for D.
Solution:
Let's assume the natural number to be x.
First condition:
When x is divided by D, the remainder is 5.
This can be represented as x ≡ 5 (mod D), where '≡' represents congruence.
Second condition:
When four times x is divided by D, the remainder is 8.
This can be represented as 4x ≡ 8 (mod D).
Using the Chinese Remainder Theorem:
Since x ≡ 5 (mod D), we can express x as x = 5 + kD, where k is a positive integer.
Substituting this value of x in the second condition, we get:
4(5 + kD) ≡ 8 (mod D)
20 + 4kD ≡ 8 (mod D)
20 ≡ 8 (mod D)
Analysis of the second condition:
The equation 20 ≡ 8 (mod D) implies that D divides (20 - 8) = 12.
So, the possible values of D are factors of 12.
Possible values for D:
The factors of 12 are 1, 2, 3, 4, 6, and 12.
Out of these, only D = 12 satisfies the first condition x ≡ 5 (mod D) because any number divisible by 12 will have a remainder of 0 when divided by 12.
Therefore, the only possible value for D is 12 (option A).
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A natural number when divided by D leaves remainder 5. When four times of the number is divided with D, we get remainder as8. Which of the following can be the value of D? (Given: D is a natural number.)a)12b)8c)6d)Both (1) and (2)e)Both (2) and (3)Correct answer is option 'A'. Can you explain this answer?
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A natural number when divided by D leaves remainder 5. When four times of the number is divided with D, we get remainder as8. Which of the following can be the value of D? (Given: D is a natural number.)a)12b)8c)6d)Both (1) and (2)e)Both (2) and (3)Correct answer is option 'A'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about A natural number when divided by D leaves remainder 5. When four times of the number is divided with D, we get remainder as8. Which of the following can be the value of D? (Given: D is a natural number.)a)12b)8c)6d)Both (1) and (2)e)Both (2) and (3)Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A natural number when divided by D leaves remainder 5. When four times of the number is divided with D, we get remainder as8. Which of the following can be the value of D? (Given: D is a natural number.)a)12b)8c)6d)Both (1) and (2)e)Both (2) and (3)Correct answer is option 'A'. Can you explain this answer?.
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