Question
Download Solution PDFFind the number when successively divided by 3,5, and 7 leaves the remainder 2,1, and 3, respectively, and the last quotient is 3.
Answer (Detailed Solution Below)
Detailed Solution
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Successive division = 3, 5, and 7
Successive remainder = 2, 1, and 3
Let the last quotient = x
then number will be
3 [5 {7x + 3} + 1] + 2
⇒ 3 [35x + 15 + 1] + 2
⇒ 105x + 45 + 3 + 2
⇒ 105x + 50
⇒ 5[21x + 10]
Since, the obtained number is a multiple of 5 then according to the option either 360 or 365 will be the correct answer.
Now, if number = 365
105x + 50 = 365
⇒ 105x = 315
⇒ x = 3
Here, we get the integer value of x, so the correct number will be 365.
∴ The correct answer is option (3).
Alternate Method
Given:
Number when successively divided by 3, 5, and 7 leaves remainders 2, 1, and 3, respectively, and the last quotient is 3.
Formula Used:
Chinese Remainder Theorem
Calculation:
Let's denote the number as ( N ).
According to the problem, when ( N ) is successively divided by 3, 5, and 7, it leaves remainders 2, 1, and 3, respectively, with the last quotient being 3.
We can break this down into a step-by-step process.
Let’s consider the successive division:
First Division by 3: ( N ) when divided by 3 leaves a remainder of 2. Thus, N = 3k + 2 for some integer k.
Second Division by 5: The quotient ( k ), obtained from the first division, when divided by 5, leaves a remainder of 1. Thus, k = 5m + 1 for some integer m.
Substituting back into the equation for ( N ): N = 3(5m + 1) + 2 = 15m + 3 + 2 = 15m + 5.
Third Division by 7: The quotient ( m ), obtained from the second division, when divided by 7, leaves a remainder of 3. Thus, m = 7n + 3 for some integer n.
Substituting back into the equation for ( N ): N = 15(7n + 3) + 5 = 105n + 45 + 5 = 105n + 50.
Last Quotient: It’s given that the last quotient from the third division is 3.
Hence, n = 3. Substituting (n = 3) into the equation for N : N = 105(3) + 50 = 315 + 50 = 365.
Thus, the number ( N ) that satisfies all the conditions is 365.
Last updated on Jun 26, 2025
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