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Find two 4 digit numbers that multiply to give 4^8 + 6^8 + 9^8

Genady
in Brain Teasers and Puzzles

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The whole puzzle is in the title. It took a friend of mine, in his own words, "3 pages of factoring (correcting and replacing)". It took me 6 lines. Take a challenge?

Would finding low prime factors be of any use?  Maybe not.  

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9 minutes ago, TheVat said:

Would finding low prime factors be of any use?  Maybe not.  

I don't know how I would do this except by going backward, i.e. after finding the answer to the puzzle looking for its prime factors.

  • Author

Step 1:

48 + 68 + 98 = 22*8 + 28*38 + 32*8 

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Step 1:           48 + 68 + 98 = 22*8 + 28*38 + 32*8 

Step 1.5:                             = (28)2 + 28*38 + (38)2 

  • Author
On 1/1/2022 at 8:11 PM, TheVat said:

Would finding low prime factors be of any use?  Maybe not.  

Did you mean these "low prime factors" -- see the previous post? If so, I apologize. My mistake, I didn't understand you. You were right.

From this point, you need only one "trick" more, and not a very uncommon one.

  • Author

Step 1:           48 + 68 + 98 = 22*8 + 28*38 + 32*8 

Step 1.5:                             = (28)2 + 28*38 + (38)2 

Step 2:                               = (28)2 + 2*28*38 + (38)2 - 28*38

  • Author

Step 1:           48 + 68 + 98 = 22*8 + 28*38 + 32*8 

Step 1.5:                             = (28)2 + 28*38 + (38)2 

Step 2:                               = (28)2 + 2*28*38 + (38)2 - 28*38

Step 2.5:                            = (28 + 38)2 - (24*34)2

  • Author

Step 1:           48 + 68 + 98 = 22*8 + 28*38 + 32*8 

Step 1.5:                             = (28)2 + 28*38 + (38)2 

Step 2:                               = (28)2 + 2*28*38 + (38)2 - 28*38

Step 2.5:                            = (28 + 38)2 - (24*34)2

Step 3:                            = (28 + 38 + 24*34)(28 + 38 - 24*34) = 8113*5521

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