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How many positive four-digit integers n have the property that the three-digit number obtained by removing the leftmost digit is one-ninth of n?

 Apr 6, 2019
 #1
avatar+101737 
+1

How many positive four-digit integers n have the property that the three-digit number obtained by removing the leftmost digit is one-ninth of n?

 

n=1000a+100b+10c+d                    where a,b,c and d are all single digit numbers.

 

n-1000a = n/9

9n-9000a=n

8n=9000a

8000a+800b+80c+8d=9000a

800b+80c+8d=1000a

100b+10c+d=125a

 

 

If a=1       

100b+10c+d=125       so the number is   1125

if a=2

100b+10c+d=250       so the number is    2250

if a=3

100b+10c+d=375       so the number is    3375

if a=4

100b+10c+d=500       so the number is    4500

if a=5

100b+10c+d=625       so the number is    5625

if a=6

100b+10c+d=750       so the number is    6750

if a=7

100b+10c+d=875       so the number is    7875

if a=8

100b+10c+d=1000       so the number is    9000    That is no good

if a=9

100b+10c+d=1125       so the number is  5 digits which is too big.

 

So there are exactly 7 four digit numbers that meet this description.

 Apr 6, 2019
edited by Melody  Apr 6, 2019
 #2
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+1

Melody: I found only 7 such numbers:

 

n=1107; m=1;if((n - 1000*m) / n ==1/9, goto3, goto4);printn;n=n+9; if(n<10000, goto2, discard=0;

 

n = 1125, 2250, 3375, 4500, 5625, 6750, 7875

 Apr 6, 2019
edited by Guest  Apr 6, 2019
 #3
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+1

The above 7 numbers are all multiples of the first one, 1125.

Guest Apr 6, 2019
 #4
avatar+101737 
0

Yes sorry.

It has just been pointed out to me that I mad a careless error that trivialized the answer. (My wrong answer I mean.)

It should not have been 8, it should have been 8d in the working.

I'll fix that and then I will think about it    frown

 Apr 6, 2019
 #5
avatar+101737 
0

My answer is correct now.   :)

Thanks guest for the little line of code to work it out.

What language is it coding for?

I must learn some coding one day, it is on my 'to do' list LOL

 

Also thanks guest for recognising that all these numbers are multiples of 1125

I can see it is logical that this would happen.      :)

Melody  Apr 6, 2019
 #6
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+1

It is in C++ language.

Guest Apr 7, 2019
 #7
avatar+101737 
0

Thanks you :)

Melody  Apr 7, 2019

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