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1 Photograph, drawing, diagram, or graph supporting the headline assertion Call-out, if needed: no more than two lines In an assertion-evidence slide, the headline is a sentence, no more than two lines, that states the slide’s assertion

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Chinese Remainder Theorem equivalence lets us use systems of congruences to solve problems Solve the system: How many solutions? –Find them. 2

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If p is prime and doesn’t divide a, then a (p-1) ≡1(mod p) This is called Fermat’s Little Theorem Examples: –2 2 =1(mod 3) –6 4 =1(mod ???) –(3 2000 )(mod 19) 3 1 2 3 … p-1 S= f(1)=a f(2)=2a f(3)=3a … f(p-1)= (p-1)a p-1 distinct values; just re-ordering of S

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Euler’s Theorem is like Fermat’s, but for composite moduli So what’s (n)? 4

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(n) is the number of integers a, such that 1 ≤ a ≤ n and gcd(a,n) = 1. Ex: (10) = 4. What’s (p) for p prime? What about (n), where n =pq (a product of 2 primes)? 5

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Xenon headlights illuminate signs better than halogen headlights do [Sylvania, 2008 ] Xenon Headlight Halogen Headlight SilverStar Ultra TM Standard Halogen Xenon

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Early detection methods can identify small tumors and therefore improve survival rates of patients [Lai et al., 2007]

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By repeating the MRI process, images from different “slices” can be stacked to create a three-dimensional image

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How much power from the engine of a semi-tractor trailer is needed to overcome aerodynamic drag? At typical highway speeds, overcoming drag requires about two-thirds of a truck engine’s output [McCallen, 2004] Aerodynamic Drag C D = 0.6 Rolling Friction and Accessories

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Lecture 2-3 Basic Number Theory and Algebra. In modern cryptographic systems, the messages are represented by numerical values prior to being encrypted.

Lecture 2-3 Basic Number Theory and Algebra. In modern cryptographic systems, the messages are represented by numerical values prior to being encrypted.

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