Proof That 10-adic Number B is Divisible by A

[tomkoolen]

The question at hand:

Let A be a 10-adic number, not a zero divisor. Proof that a 10-adic number B is dividible by A if 2^q*5^p*B has ends with p+q zeroes.

My work so far:

Because A is not a zero divisor, it is not dividible by all powers of 2 nor 5, so it follows from a theorem that A = 2^q*5^p*C with C invertible and p and q natural numbers. Now I have no clue how to connect this with 0. If anybody could help me out, I would be very grateful.

Thanks in advance.

 

[jvicens]

Hello,

Great question! Let's start by breaking down the problem into smaller parts. First, we know that A is a 10-adic number, meaning it can be represented as a series of digits in base 10 with infinitely many digits to the left of the decimal point. Second, we know that A is not a zero divisor, meaning it is not equal to zero and does not multiply to give zero.

Next, you correctly stated that A can be written as A = 2^q*5^p*C, where C is an invertible 10-adic number and p and q are natural numbers. This means that A has a finite number of digits to the left of the decimal point, specifically p digits of 5 and q digits of 2.

Now, let's look at the number B. We know that B is also a 10-adic number, and we want to show that it is divisible by A. We are given that 2^q*5^p*B has p+q zeroes at the end. This means that B has at least p+q digits of 2 and 5 combined to the right of the decimal point.

To show that B is divisible by A, we can rewrite B as B = 2^q*5^p*C'*D, where C' is an invertible 10-adic number and D is the remaining digits of B. Notice that D has at least p+q digits to the right of the decimal point, which means it can be written as D = 2^(p+q)*E, where E is a 10-adic number.

Now, let's look at the product 2^q*5^p*B. This can be rewritten as 2^q*5^p*2^q*5^p*C'*D, which simplifies to 2^(2q)*5^(2p)*C'*E. Notice that the product ends with p+q zeroes, just like we were given in the problem.

Since C' is an invertible 10-adic number, we can divide both sides by it, giving us 2^(2q)*5^(2p)*E = A*B. This shows that A divides B, as desired.

I hope this helps to connect the given information with the concept of zero divisors and to prove that A divides B. Let me know