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The integers 34041 and 32506,when divided by a three digit integer N, leave the same remainder. What can be the value of N?
- a)289
- b)307
- c)317
- d)319
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The integers 34041 and 32506,when divided by a three digit integer N, ...
Let the common remainder be x. Then numbers (34041 – x) and (32506 – x) would be completely divisible by n. Hence the difference of the numbers (34041 – x) and (32506 – x) will also be divisible by n or (34041 – x – 32506 + x) = 1535 will also be divisible by n. Now, using options we find that 1535 is divisible by 307.
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Most Upvoted Answer
The integers 34041 and 32506,when divided by a three digit integer N, ...
Find the difference between both 34041 and 32506.
34041 - 32506 = 1535
So N should be a factor of 1535.
Factors of 1535:-
1, 5, 307, 1535
Here, 307 is a three digit factor of 1535. So N = 307
34041 and 32506 divided by 307 leaves remainder 271.
So 307 is the answer.
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Community Answer
The integers 34041 and 32506,when divided by a three digit integer N, ...
Given:
- Integers 34041 and 32506 leave the same remainder when divided by a three digit integer N.
To find:
- The possible value(s) of N.
Solution:
- Let the common remainder be 'r'.
- Then, we can write:
- 34041 = kN + r ...(1)
- 32506 = mN + r ...(2)
where k and m are some integers.
- Subtracting equation (2) from (1), we get:
- 1535 = (k - m)N
- As N is a three digit integer, its factors are also three digit integers.
- Factors of 1535 are 5, 307, and 1535 itself.
- As N is greater than the remainder 'r', it must be greater than or equal to 1 and less than or equal to 512 (the largest possible value of 'r' is 511).
- Testing each of the factors of 1535 in this range, we get:
- When N = 307, we have:
- k = 111, r = 24
- m = 107, r = 24
So, N = 307 satisfies the given conditions.
- When N = 5 or 1535, we have:
- k and m cannot be integers, as N is a factor of their difference.
So, N = 5 and N = 1535 do not satisfy the given conditions.
- Therefore, the possible value of N is 307.
Answer:
- The possible value of N is (B) 307.
- Integers 34041 and 32506 leave the same remainder when divided by a three digit integer N.
To find:
- The possible value(s) of N.
Solution:
- Let the common remainder be 'r'.
- Then, we can write:
- 34041 = kN + r ...(1)
- 32506 = mN + r ...(2)
where k and m are some integers.
- Subtracting equation (2) from (1), we get:
- 1535 = (k - m)N
- As N is a three digit integer, its factors are also three digit integers.
- Factors of 1535 are 5, 307, and 1535 itself.
- As N is greater than the remainder 'r', it must be greater than or equal to 1 and less than or equal to 512 (the largest possible value of 'r' is 511).
- Testing each of the factors of 1535 in this range, we get:
- When N = 307, we have:
- k = 111, r = 24
- m = 107, r = 24
So, N = 307 satisfies the given conditions.
- When N = 5 or 1535, we have:
- k and m cannot be integers, as N is a factor of their difference.
So, N = 5 and N = 1535 do not satisfy the given conditions.
- Therefore, the possible value of N is 307.
Answer:
- The possible value of N is (B) 307.
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The integers 34041 and 32506,when divided by a three digit integer N, leave the same remainder. What can be the value of N?a)289b)307c)317d)319Correct answer is option 'B'. Can you explain this answer?
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