SOLUTION: When a certain four-digit number is multiplied by 4 its digits appear in reverse order. It also has the following properties. Its first digit is a quarter of the last one. And its

Question 436640: When a certain four-digit number is multiplied by 4 its digits appear in reverse order. It also has the following properties. Its first digit is a quarter of the last one. And its second digit is one less than the first. What number must it be?
Found 2 solutions by ankor@dixie-net.com, Edwin McCravy:
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
When a certain four-digit number is multiplied by 4 its digits appear in reverse order.
It also has the following properties. Its first digit is a quarter of the last one.
And its second digit is one less than the first. What number must it be?
:
Let the four digit number = 1000a + 100b + 10c + d
:
"When a four-digit number is multiplied by 4 its digits appear in reverse order."
4(1000a+100b+10c+d) = 1000d + 100c + 10b + a
:
4000a + 400b + 40c + 4d = 1000d + 100c + 10b + a
4000a - a + 400b - 10b = 100c - 40c + 1000d - 4d
3999a + 390b = 60c + 996d
Simplify, divide by 3
1333a + 130b = 20c + 332d
:
" Its first digit is a quarter of the last one."
a = .25d
or
d = 4a
:
"its second digit is one less than the first.
b = (a-1)
:
1333a + 130b = 20c + 332d
Replace b and d
1333a + 130(a-1) = 20c + 332(4a)
1333a + 130a - 130 = 20c + 1328a
1333a + 130a - 1328a = 20c + 130
135a = 20c + 130
Simplify, divide by 5
27a = 4c + 26
a = c +
A close inspection of this equatiion reveals there is only one single digit integer solution
c = 7; a = 2
:
find b and d
b = 2 - 1
b = 1
and
d = 4(2)
d = 8
:
2178 is the number
:
Check this: 4 * 2178 = 8712
Answer by Edwin McCravy(20056) (Show Source): You can put this solution on YOUR website!
2178 x 4 = 8712
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