## Units digit of n - DS (>700)

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### Units digit of n - DS (>700)

If $(243)^x(463)^y = n$, what is the units digit of n?

(1) $x + y = 7$
(2) $x = 4$
respuesta: show

Luke
Mensajes: 258

### Re: Units digit of n - DS (>700)

La conclusión de que (2) is not sufficient:

If x=4, that means units digit of (243)^4=1. However, we don't know anything about y. If y is odd, units digits of n can be 3 or 7; if y is even, units digits of n can be 9 or 1. Then, we can only say digit unit of n is odd, but we can have four different values: 3,7,9 or 1, depending on the value of y - 2 is not sufficient

La conclusión de que (1) is sufficient:

We know both x and y can have four possible values each one: 3 (x or y=1), 9 (x or y=2), 7 (x or y=3) and 1 (x or y=4) (the sequence is repeated for lower/higher values of x and y).

If x+y=7, we can have infinite combinations of the pair (x,y):
-2,9
-1,8
0,7
1,6
2,5
etc.
However, this fixes the possible relation between (243)^x and (463)^y! We can see this if we write down the sequence in a wide range of values for x and y:

Power Unit digits
-3 3
-2 9
-1 7
0 1
1 3
2 9
3 7
4 1
5 3
6 9
7 7
8 1
9 3
10 9

If x=-2, y must be equal to 9. In that case, unit digit of n is equal to 7*3=21
We can see every combination of x+y=7 yields a result of 7, so (1) is sufficient.

I hope is clearly explained!!
I took me more than 2 minutes and my first choice was (3) Together

Luke
Mensajes: 258

### Re: Units digit of n - DS (>700)

Ufffff!!! you are right, what a great assumption!!

Yes I am :-(

lox
Mensajes: 106

### Re: Units digit of n - DS (>700)

Las unidades de un producto siempre son el producto de las unidades de todos los numeros que estemos multiplicando, en este caso el producto de 3 "x veces" por 3 "y veces", es decir $3^x*3^y = 3^{x+y}$ y si sabemos lo que vale x + y esta hecho.

Luke