document.write( "Question 1207737: Carlos wants to send Cecil a message encrypted with RSA. Cecil has published his public encryption exponent $4$ and his public modulus $15$. When Carlos encrypts the number $11$ with this system, what is the result? \n" ); document.write( "
Algebra.Com's Answer #845880 by math_tutor2020(3816)\"\" \"About 
You can put this solution on YOUR website!

\n" ); document.write( "Tutor Edwin has a great solution.
\n" ); document.write( "I'll show another way to compute 11^4 (mod 15).
\n" ); document.write( "First let's determine 11^2 in mod 15.\r
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\n" ); document.write( "\n" ); document.write( "11^2 = 121 = 1 (mod 15)\r
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\n" ); document.write( "\n" ); document.write( "How did I go from 121 to 1?
\n" ); document.write( "Well notice that 121/15 = 8 remainder 1
\n" ); document.write( "Ignore the quotient. All we care about is the remainder with modular arithmetic.
\n" ); document.write( "Both 121 and 1 have the same remainder when dividing over 15.\r
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\n" ); document.write( "\n" ); document.write( "So that's how 121 = 1 (mod 15)\r
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\n" ); document.write( "\n" ); document.write( "Side note: If a = b (mod n), then a-b is a multiple of n. In other words, a-b = nk for some integer k.
\n" ); document.write( "In the claim above we have 121-1 = 120 as a multiple of 15 (since 15*8 = 120).\r
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\n" ); document.write( "\n" ); document.write( "Then,
\n" ); document.write( "11^2 = 1 (mod 15)
\n" ); document.write( "(11^2)^2 = 1^2 (mod 15)
\n" ); document.write( "11^(2*2) = 1 (mod 15)
\n" ); document.write( "11^4 = 1 (mod 15)\r
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\n" ); document.write( "\n" ); document.write( "Dividing 11^4 over 15 gives some quotient with remainder 1.
\n" ); document.write( "You can use long division to check this claim as Edwin has done.
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