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A number when divided by 7 leaves a remainder 3 and the resulting quotient when divided by 11 leaves a reminder 6 if the same number when divided by 11 leaves a reminder m and the resulting quotient when divided by 7 leaves a reminder n what are the value of m and n respectively?
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A number when divided by 7 leaves a remainder 3 and the resulting quot...
The given information can be represented in the form of congruences:
x ≡ 3 (mod 7) and (x/7) ≡ 6 (mod 11)
The first congruence states that when the number x is divided by 7, the remainder is 3. This can be written as x = 7k + 3 for some integer k.
Substituting this value of x in the second congruence, we get:
(7k + 3) / 7 ≡ 6 (mod 11)
Simplifying the above congruence we get
k + 3/7 ≡ 6 (mod 11)
Now the question is asking for the remainder when the number is divided by 11 and the quotient when divided by 7.
so when the same number x = 7k + 3 is divided by 11 we get:
x ≡ 3 (mod 11)
and the quotient when divided by 7 is k, so we have:
x/7 = k (mod 7)
Therefore, m = 3 and n = k.
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Most Upvoted Answer
A number when divided by 7 leaves a remainder 3 and the resulting quot...
Understanding the Problem
We have a number, let's denote it as x. The problem states the following conditions:
- When x is divided by 7, it leaves a remainder of 3.
- The quotient from the first division when divided by 11 leaves a remainder of 6.
Finding the Number x
From the first condition, we can express x as:
- x = 7k + 3 (for some integer k)
When we divide x by 7, the quotient is:
- Quotient = k
Now applying the second condition:
- k when divided by 11 leaves a remainder of 6, so we can write:
- k = 11j + 6 (for some integer j)
Substituting this back into our equation for x, we get:
- x = 7(11j + 6) + 3 = 77j + 42 + 3 = 77j + 45
Thus, the general form of x is:
- x = 77j + 45
Finding m and n
Now, to find the values of m and n, we will check how x behaves when divided by 11 and 7 respectively.
1. Finding m: x mod 11
- x = 77j + 45
- Since 77 is a multiple of 11, we have:
- x mod 11 = 45 mod 11 = 1
- Therefore, m = 1.
2. Finding n: Quotient of x divided by 11 mod 7
- Quotient = (77j + 45) / 11 = 7j + 4 (ignoring the remainder)
- Now, 7j + 4 mod 7 = 4
- Thus, n = 4.
Final Values
- m = 1
- n = 4
We have a number, let's denote it as x. The problem states the following conditions:
- When x is divided by 7, it leaves a remainder of 3.
- The quotient from the first division when divided by 11 leaves a remainder of 6.
Finding the Number x
From the first condition, we can express x as:
- x = 7k + 3 (for some integer k)
When we divide x by 7, the quotient is:
- Quotient = k
Now applying the second condition:
- k when divided by 11 leaves a remainder of 6, so we can write:
- k = 11j + 6 (for some integer j)
Substituting this back into our equation for x, we get:
- x = 7(11j + 6) + 3 = 77j + 42 + 3 = 77j + 45
Thus, the general form of x is:
- x = 77j + 45
Finding m and n
Now, to find the values of m and n, we will check how x behaves when divided by 11 and 7 respectively.
1. Finding m: x mod 11
- x = 77j + 45
- Since 77 is a multiple of 11, we have:
- x mod 11 = 45 mod 11 = 1
- Therefore, m = 1.
2. Finding n: Quotient of x divided by 11 mod 7
- Quotient = (77j + 45) / 11 = 7j + 4 (ignoring the remainder)
- Now, 7j + 4 mod 7 = 4
- Thus, n = 4.
Final Values
- m = 1
- n = 4
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A number when divided by 7 leaves a remainder 3 and the resulting quotient when divided by 11 leaves a reminder 6 if the same number when divided by 11 leaves a reminder m and the resulting quotient when divided by 7 leaves a reminder n what are the value of m and n respectively? for Class 9 2025 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about A number when divided by 7 leaves a remainder 3 and the resulting quotient when divided by 11 leaves a reminder 6 if the same number when divided by 11 leaves a reminder m and the resulting quotient when divided by 7 leaves a reminder n what are the value of m and n respectively? covers all topics & solutions for Class 9 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A number when divided by 7 leaves a remainder 3 and the resulting quotient when divided by 11 leaves a reminder 6 if the same number when divided by 11 leaves a reminder m and the resulting quotient when divided by 7 leaves a reminder n what are the value of m and n respectively?.
A number when divided by 7 leaves a remainder 3 and the resulting quotient when divided by 11 leaves a reminder 6 if the same number when divided by 11 leaves a reminder m and the resulting quotient when divided by 7 leaves a reminder n what are the value of m and n respectively? for Class 9 2025 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about A number when divided by 7 leaves a remainder 3 and the resulting quotient when divided by 11 leaves a reminder 6 if the same number when divided by 11 leaves a reminder m and the resulting quotient when divided by 7 leaves a reminder n what are the value of m and n respectively? covers all topics & solutions for Class 9 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A number when divided by 7 leaves a remainder 3 and the resulting quotient when divided by 11 leaves a reminder 6 if the same number when divided by 11 leaves a reminder m and the resulting quotient when divided by 7 leaves a reminder n what are the value of m and n respectively?.
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