# To find the remainders.

A number when divided successively by $4$ and $5$ leaves remainder $1$ and $4$ respectively. When it is divided by $5$ and $4$, then the respective remainders are:

$1,2$

$2,3$

$3,2$

$4,1$

I thought of picking a number, say $9$, which satisfies the given conditions, and the answer it gave me was option $D$. But my textbook states that answer is $B$ (They have taken $37$ as their number ). I do not understand why they have taken number 37.

Is there any method for this ?

• I think this needs clarification. What's the difference between "dividing successively by 4 and 5 " and "dividing successively by 5 and 4"? Any interpretation of those phrases I can think of would just yield the same remainders in reverse order. – lulu Jul 22 '15 at 11:50
• I think it means first by 4 then by 5.. – MonK Jul 22 '15 at 11:50
• @MonK yes this is what it means – Taylor Ted Jul 22 '15 at 11:50
• @MonK No that is not .see answer below of user lab bhattacharjee – Taylor Ted Jul 22 '15 at 11:52

$$N=4a+1, a=5b+4\implies N=4(5b+4)+1=20b+17$$

$$N=5(4b+3)+2, 4b+3=4(b)+3$$

• why is a =5b +4 – Taylor Ted Jul 22 '15 at 11:49
• @JPG, As "leaves remainder" $4$ when divided by $5$ – lab bhattacharjee Jul 22 '15 at 11:49
• shouldn't it be N=5b + 4 – Taylor Ted Jul 22 '15 at 11:50
• @JPG, What do you mean by "successively"? – lab bhattacharjee Jul 22 '15 at 11:50
• I am thinking of dividing the number first by 4 and then by 5 – Taylor Ted Jul 22 '15 at 11:51

Lets assume number is n

So n is of the form $4k+1$.

and k is of the form $5p+4$.

So $n=4(5p+4)+1=20p+16+1=20p+17$.

n could be 37,57,77 etc, starting from submitting p=1,2,3 and so on.

Now, rewrite $20p+17$ as $15p+5p+15+2=15(p+1)+5p+2$.

Clearly, if you divide this by 5 the remainder is $2$, and quotient will turn out $3(p+1)+p=3p+3+p=4p+3$

giving us a remainder of $3$ when divided by 4.

Answer is $2,3$