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?

Badada Apr 6, 2019

#1**+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.**

Melody Apr 6, 2019

#2**+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**

Guest Apr 6, 2019

edited by
Guest
Apr 6, 2019

#4**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

Melody Apr 6, 2019

#5**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