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Note that

1111111=10^6+10^5+10^4+10^3+10^2+10+1

and

909091=10^6-10^5+10^4-10^3+10^2-10+1.

 

Compute 1111111^2 - 909091^2.

 May 29, 2021
 #1
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+1

$(1111111+909091)(1111111-909091)=(2\cdot 10^6+2\cdot 10^4+2\cdot 10^2+2\cdot 10^0)(2\cdot 10^5+2\cdot 10^3+2\cdot 10)=4(10^6+10^4+10^2+10^0)(10^5+10^3+10)=4(10^{11}+10^{9}+10^{7}+10^9+10^7+10^5+10^7+10^5+10^3+10^5+10^3+10)=4(10^{11}+2\cdot 10^9+2\cdot 10^7+2\cdot 10^5+2\cdot 10^3+10)=4(102020202010)=408080808040$.

 May 30, 2021

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