Question

When a number is successively divided by $7,5$ and $4$, it leaves remainders of $4,2$ and $3$ respectively. What will be the respective remainders when the smallest such number is successively divided by $8,5$ and $6$ ?

• A. $5,0,3$
• B. $2,2,4$
• C. $3,0,3$
• D. $2,4,2$

Answer 1

Answer: (C)

$(3,0,3)$

We will first find the smallest number such that when it will be successively divided by 7, 5 and 4, it leaves remainders of 4,2 and 3 respectively.
Let the smallest number be s. Let us denote the quotient as x when s is divided by 7 leaving remainder 4 , denote the quotient as y when x is divided by 5 leaving remainder 2 and denote the quotient as z when y is divided by 4 leaving remainder 3.
By Using Euclid’s division lemma and have,
$s=7x+4$....(1)
$x=5y+2$....(2)
$y=4z+3$.....(3)

We put equation (2) in equation (1), we get
$s=7x+4$
$\Rightarrow s=7(5y+2)+4$
$\Rightarrow s=35y+18$
We put the value of y from (3) in above step to have,

$\Rightarrow s=35(4z+3)+18$
$\Rightarrow s=140z+105+18$
$\Rightarrow s=140z+123$
We see from the above equation that the smallest positive integral value of s occurs when z=0 which means s=123.
We verify by dividing 123 by 7, 5 and 4 successively to get remainders 4, 2 and 3 respectively with successive quotients 17, 3 and 3.
We are asked in the question to find the remainders when divided by 8, 5 and 6 successively. So we divide 123 by 8 to get the remainder 3 and first quotient 15. We divide the first quotient 15 by 5 to get second quotient 3 and remainder 0. We divide the second quotient 3 by 6, we get remainder 3.
So the remainders are 3,0,3 and
hence the correct option is C.