Answer to Question #228290 in Combinatorics | Number Theory for Stone

Question

The remainder of n7 is 4. The remainder of m9 is 7. Which of the following is a potential value for n×m?

(A) 24

(B) 44

(C)22

(D) 126

(E) 55

Answer 1

Let $n=7x+4, x=0,1,2,...$ and let $m=9y+7,y=0,1,2,...$ Then

$$n\times m=(7x+4)(9y+7)=63xy+49x+36y+28$$

$y=0,$ if $n\times m$ is divided by $7$ the remainder is $0.$

$$126\div7=18$$

$$m=9(0)+7=7, n=7(2)+4=18$$

$y=1,$ if $n\times m$ is divided by $7$ the remainder is $1.$

$$22\div7=3, remainder \ 1$$

$$m=9(1)+7=16$$

But $22$ is not divisible by $16.$

$y=2,$ if $n\times m$ is divided by $7$ the remainder is $2.$

$$44\div7=6, remainder \ 2$$

$$m=9(2)+7=25$$

But $44$ is not divisible by $25.$

$y=3,$ if $n\times m$ is divided by $7$ the remainder is $3.$

$$24\div7=2, remainder \ 3$$

$$m=9(3)+7=34$$

But $24$ is not divisible by $34.$

$y=4,$ if $n\times m$ is divided by $7$ the remainder is $4.$

$$44\div7=6, remainder \ 2$$

$$m=9(2)+7=25$$

But $44$ is not divisible by $25.$

$$55\div7=7, remainder \ 6$$

$y=6,$ if $n\times m$ is divided by $7$ the remainder is $6.$

$$m=9(6)+7=61$$

But $55$ is not divisible by $61.$

(D) 126