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Subject: "Digitla Roots"     Previous Topic | Next Topic
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Conferences The CTK Exchange High school Topic #403
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Sandy
guest
Feb-14-11, 04:02 PM (EST)
 
"Digitla Roots"
 
   Digital Root of a Product

On the page

https://www.cut-the-knot.org/arithmetic/rapid/defensive.shtml

Alexander casually stated

"The important fact about digital roots is that the digital root of the product of integers equals the product of their digital roots (downsized to a digital root if necessary.)"

I can't see an easy way to prove this. Can somebody help?

Thanks!


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alexbadmin
Charter Member
2767 posts
Feb-15-11, 04:29 PM (EST)
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1. "RE: Digitla Roots"
In response to message #0
 
   This is because any power of 10 modulo 9 equals 1, implying that the digital root of a number is simply the remainder of division of that number by 9.

Let you have two numbers (9a + b) and (9c + d), b being the digital root of one, d of the other. Then

(9a + b)(9c + d) + 9(ad + bc + 9ac) + bd = 9k + bd.

It follows that the remainder of the division by 9 of the product equal the remainder of the division by 9 of the remainders. But the remainders are the digital roots.


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