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Question

If three times the larger of the two numbers is divided by the smaller one, we get 4 as quotient and 3 as the remainder. Also, if seven times the smaller number is divided by the larger one, we get 5 as quotient and 1 as the remainder. Find the numbers.

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Solution

Let the larger number be x and the smaller one be y. We know that

Dividend =(Divisor × Quotient) +Remainder. (i)

When 3x is divided by y, we get 4 as quotient and 3 as remainder. Therefore, by using (i), we get

Dividend =(Divisor × Quotient) +Remainder. (i)

When 3x is divided by y, we get 4 as quotient and 3 as remainder. Therefore, by using (i), we get

3x=4y+3⇒3x−4y−3=0 (ii)

When 7y is divided by x, we get 5 as quotient and 1 as remainder. Therefore, by using (i), we get

7y=5x+1⇒5x−7y+1=0 .(iii)

Solving equations (ii) and (iii), by cross-multiplication, we get

x−4−21=−y3+15=1−21+20

x−25=−1, −y18=−1

⇒x=25 and y=18

Hence, the required numbers are 25 and 18.

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