Multiplying a Number by 4

Date: 01/11/99 at 07:48:56
From: Randy Peh
Subject: A no. x 4 = no. backwards

Dr. Math:

My math teacher told me to find a few numbers that when multiplied by 
4, will give the original number backwards. Can you help me out?

Date: 01/11/99 at 13:44:56
From: Doctor Rob
Subject: Re: A no. x 4 = no. backwards

Thanks for writing to Ask Dr. Math!

There aren't any except 0 with 1, 2, or 3 digits.

If there are four digits, there is a unique solution. First see that
the first digit of the number has to be 1 or 2. Numbers can't start
with 0, and if the first digit is 3 or more, 4 times the number will
be a five-digit number. Then the last digit of the product is 1 or 2,
but being even, it must be 2. Thus the number begins with 2. 

Now think about the last digit of the number. When multiplied by 4, it 
ends in 2, so it must be 3 or 8. Then the first digit of the product 
must be 3 or 8. But the product is 4 times a number bigger than 2000, 
so must be bigger than 8000, so the first digit of the product can't 
be 3. That means that the number must look like "2 A B 8" and four 
times it must be "8 B A 2". When you do the multiplication, the carry 
from the units to the tens column is a 3, and the carry from the 
hundreds to the thousands column is a 0 (why?). That means that 
hundreds digit A of the number can't be very large, since 4 times it 
plus the carry must be at most 9. On the other hand, A equals the tens 
digit of the product, so it must be odd (it's the last digit of 
4*B + 3). That will tell you A, and the rest is pretty easy.

If there are five digits, there is likewise one solution found by
inserting one extra digit in the middle of the number and product.  
From the answers for four and five digits, you might be able to guess 
solutions for six, seven, or more digits.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/