A four-digit integer, WXYZ, in which W,X,Y and Z each represent a different digit, is formed according to the?
A four-digit integer, WXYZ, is formed according to the following rules:
- X = W + Y + Z^2
- W = Y + 13
- Z = W - 5
In this context, W, X, Y, and Z each represent different digits. How can we determine the values of W, X, Y, and Z based on these equations?
4 Answers
Feb 21, 2025
W is the key variable
Z = W-5
Y = W-1
X = W+Y+Z = W+(W-1)+ (W-5) = 3W-6
This means:
W can't be less than 5 (or W-5 < 0)
W can't be greater than 5 (or 3W - 6 > 9)
So WXYZ = 5940 is the only solution.
Feb 22, 2025
from 3) W = Z + 5, so W is at least 5. can W be more than 5? let's see what that means.
if W = 6, then Y = 5, and W + Y + Z is not a single digit number. it only gets worse if W is bigger.
so W must be 5, in which case, Y = 4, Z = 0 and X = 5+4+0 = 9.
the number is thus
5940
you can rewrite 2 as y=w-1
this means that w+w-1+w-5=x
since they are all single digit numbers then w+w-1+w-5<=9
and since z is the smallest z>=0
if 3w-6=x then x must be a multiple of 3
w>=5 since z>=0
w<=5 since 6*3-5>9
therefore w=5 now all the rest are easy
w=5, x=9, y=4, z=0
the integer is 5940
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