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A certain number when successfully divided by 3 5 7 leaves remainders 1 2 3 respectively what is the complete remainder when the same number is divided by 105?
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A certain number when successfully divided by 3 5 7 leaves remainders ...
Explanation:
Given Information:
- Number leaves remainder 1 when divided by 3
- Number leaves remainder 2 when divided by 5
- Number leaves remainder 3 when divided by 7
Finding the Number:
To find the number, we can use the Chinese Remainder Theorem (CRT). The CRT states that if we have a system of congruences of the form:
\[x \equiv a_1 \pmod{m_1}, x \equiv a_2 \pmod{m_2}, \ldots, x \equiv a_n \pmod{m_n}\]
Where \(m_1, m_2, \ldots, m_n\) are pairwise coprime (i.e., gcd of any two numbers is 1), then there exists a unique solution modulo \(m_1 \times m_2 \times \ldots \times m_n\).
Applying the CRT to the given information:
\[x \equiv 1 \pmod{3}, x \equiv 2 \pmod{5}, x \equiv 3 \pmod{7}\]
We get:
\[x \equiv 23 \pmod{105}\]
Complete Remainder when divided by 105:
When the number is divided by 105, the complete remainder will be 23. This is because the remainder obtained when dividing by 105 is the same as the remainder obtained when dividing by 23.
Therefore, the complete remainder when the number is divided by 105 is 23.
Given Information:
- Number leaves remainder 1 when divided by 3
- Number leaves remainder 2 when divided by 5
- Number leaves remainder 3 when divided by 7
Finding the Number:
To find the number, we can use the Chinese Remainder Theorem (CRT). The CRT states that if we have a system of congruences of the form:
\[x \equiv a_1 \pmod{m_1}, x \equiv a_2 \pmod{m_2}, \ldots, x \equiv a_n \pmod{m_n}\]
Where \(m_1, m_2, \ldots, m_n\) are pairwise coprime (i.e., gcd of any two numbers is 1), then there exists a unique solution modulo \(m_1 \times m_2 \times \ldots \times m_n\).
Applying the CRT to the given information:
\[x \equiv 1 \pmod{3}, x \equiv 2 \pmod{5}, x \equiv 3 \pmod{7}\]
We get:
\[x \equiv 23 \pmod{105}\]
Complete Remainder when divided by 105:
When the number is divided by 105, the complete remainder will be 23. This is because the remainder obtained when dividing by 105 is the same as the remainder obtained when dividing by 23.
Therefore, the complete remainder when the number is divided by 105 is 23.
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A certain number when successfully divided by 3 5 7 leaves remainders 1 2 3 respectively what is the complete remainder when the same number is divided by 105?
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A certain number when successfully divided by 3 5 7 leaves remainders 1 2 3 respectively what is the complete remainder when the same number is divided by 105? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A certain number when successfully divided by 3 5 7 leaves remainders 1 2 3 respectively what is the complete remainder when the same number is divided by 105? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A certain number when successfully divided by 3 5 7 leaves remainders 1 2 3 respectively what is the complete remainder when the same number is divided by 105?.
A certain number when successfully divided by 3 5 7 leaves remainders 1 2 3 respectively what is the complete remainder when the same number is divided by 105? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A certain number when successfully divided by 3 5 7 leaves remainders 1 2 3 respectively what is the complete remainder when the same number is divided by 105? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A certain number when successfully divided by 3 5 7 leaves remainders 1 2 3 respectively what is the complete remainder when the same number is divided by 105?.
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