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# How many terminal zeroes does 40^8 * 75^18 have?

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How many terminal zeroes does 40^8 * 75^18 have?

Mar 24, 2019

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How many terminal zeroes does 40^8 * 75^18 have?

Well, 40 to the 8th will have 8 terminal zeros.  However many terminal zeros in the base, there will be that many times the exponent in the answer.      examples:  102=100  103=1,000  104=10,000  1002=10,000  1,0002=1,000,000

As regards the second factor, any number ending with 5 no matter to what exponent you raise it, the answer will always end with a 5.     examples: 52=25  152=225  252=625  1052=11,025  etc.  So the second factor adds no terminal zeros to the product.

Multiplying the two factors together, there are 8 terminal zeros in the product.

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Mar 24, 2019
edited by Guest  Mar 24, 2019
#2
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Note that we can write  40^8 * 75^18  as

( 4 * 10)^8  * ( 3 * 5^2)^18  =

( 2^3 * 5)^8 *  ( 3 * 5^2)^18  =

(2 ^3)^8 * (5)^8  * 5^36 * 3 ^18 =

2^24 * 5^44 * 3^18  =

2^24 * 5^24  * 5^20 * 3^18

(2 * 5)^24  *  5^20 * 3^18  =

5^20 * 3^18  *  ( 2 * 5) ^24

[5^20 * 3^18  ]  * (10)^24       which   has  24  terminal zeroes   Mar 24, 2019
edited by CPhill  Mar 24, 2019
#3
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I'm the Guest who posted the first answer.  I see how I made a mistake.  408 is 65,536 x 108

The 6 at the end of 65,536 times the 5 at the end of 7518 will produce at least one additional terminal zero.

I need go no further than that to realize the approach to my answer was simplistic, and, worse, it was wrong.

I'm mortified to make such an elementary arithmetic oversight, and I apologize if my blunder led anyone astray.

Mar 24, 2019
edited by Guest  Mar 24, 2019
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LOL!!!!....don't worry about it, Guest....we all make mistakes on here!!!

[ I doubt that anyone will press charges against you   .....  ]   CPhill  Mar 24, 2019