# Find digits $x$, $y$, $z$, and $p$ such that $(xyzp)_{10}=4\cdot (pzyx)_{10}$

Given that equation $xyzp=4\cdot pzyx$ is valid, how do I solve showing my work clearly? By trial and error the valid answers are $x=8,y=7,z=1,p=2$. These are the only conditions provided.

-
Is everything needed in your question? – Amzoti Jul 7 '13 at 4:35
Does your question mean, "how to solve the problem without trial and error?" – フラン Jul 7 '13 at 4:38
I suspect that the variables are supposed to be digits (not positive reals), and we are talking about concatentation rather than multiplication. – Gerry Myerson Jul 7 '13 at 4:41
@Gerry: Which makes the $4$ even more confusing... – Zev Chonoles Jul 7 '13 at 4:42
This question appears to be off-topic because "If and only if the (whys) are answered because here take it I never gave out my answers." – Will Jagy Jul 7 '13 at 5:14

Here's a start (assuming I have the right interpretation of the problem in the comments).

1. $p$ must be even (why?)

2. $p\lt3$ (why?)

3. So $x$ is 8 or 9 (why?)

4. But $x$ can't be 9 (why?)

-
If and only if the (whys) are answered because here take it I never gave out my answers. – DOCTOR NGILAZI BANDA JOSHUA Jul 7 '13 at 4:57
I'm sorry --- I have no idea what you mean by that. – Gerry Myerson Jul 7 '13 at 5:57

p z y x

## × 4

x y z p x × 4 = p in unit place. similarly p × 4 = x

4 digit number × 4 gives 4 digit answer, p < 3 But p = 4 × x is even so p = 2 and x = 8 so it becomes 2 z y 8

## × 4

8 y z 2 it shows that there is no carry forward in z × 4 so z must be 1 2 1 y 8

## × 4

8 y 1 2 so you can calsukate y 8 × 4 = 32 ----- 3 c/f. y × 4 + 3 = 1 in unit place so y = 7

-