Divisibility by 7

[chitrangada.maitra]

This problem has been troubling me for a while.
Strategy Guide: Number Properties, page 134, #20

x has a remainder of 5 when divided by 9. y has a remainder of 7 when divided by 9. what is the remainder when x-y is divided by 9

My Approach:
X can be 5, 14, 23, 32, 41... etc
Y can be 7, 16, 25, 34, 43... etc

If X>y, the remainder of X-Y upon division by 9 is consistently 7.
However, if, X<Y, X-Y can be -2, -11, -20.... etc
These numbers upon division by 9 does NOT give one consistent answer.

The strategy guide suggests adding the divisor to a negative remainder -- that is the part that confuses me.

Can someone please suggest a simpler way to solve this.

Thanks,

[adiagr]

This problem has been troubling me for a while.
Strategy Guide: Number Properties, page 134, #20

x has a remainder of 5 when divided by 9. y has a remainder of 7 when divided by 9. what is the remainder when x-y is divided by 9

My Approach:
X can be 5, 14, 23, 32, 41... etc
Y can be 7, 16, 25, 34, 43... etc

If X>y, the remainder of X-Y upon division by 9 is consistently 7.
However, if, X<Y, X-Y can be -2, -11, -20.... etc
These numbers upon division by 9 does NOT give one consistent answer.

The strategy guide suggests adding the divisor to a negative remainder -- that is the part that confuses me.

Can someone please suggest a simpler way to solve this.

Thanks,

Algebra wont be a bad option in this case.

x = 9 k + 5
y = 9z+5

k, z any integer.

x-y = 9(k-z)- 2

so when x-y is divided by 9, we see remainder is (-2)

add divisor to (-)2

9-2 = 7 ---> this will be the remainder.

Aditya

[tim]

technically, -2, -11, -20 etc have a remainder of 7 when divided by 9. the only thing you have to do is, as suggested, add or subtract the divisor (in this case 9) until you get something in the range of 0 to 8. this is what you should do regardless of whether x-y is -20 or 34. if you want to avoid the hassle though, just pick your own values of x and y that make you happy.. :)

[dlbrownct]

OKay, I figured out the -2 part pretty quickly. Thus the actual remainder is -2/9 correct? However, I'm not 'getting' the part of adding the divisor back to the -2 to get 7.

x=9m+5
y=9n+7

x-y=9m+5-9n-7 = 9(m-n)-2
Now, (x-y)/9 has the remainder of -2, right?

Can you please clarify my misunderstanding? I know I am real close, but not close enough.
Thanks

Edit: The remainder of a division problem cannot be negative. Makes perfect sense of course. :)

[jnelson0612]

dlbrown, it seems that you've got it now. Let us know if you need more guidance.