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122b a p21 1 (modulo p) if a is not divisible by p P Proof: Expand (x 1) p by the Binomial Theorem (x 1) p x p 1 p 1 p xi i 1 i p and note that the binomial coef cient i is divisible by p for 1 i p 2 1 This proves (x 1) p (1 x p) (modulo p) so that Proposition 122a follows by induction Writing a p 2 a a(a p21 2 1) 0 (modulo p) and assuming gcd{a, p} 1, both sides may be divided by a (equivalently multiplied by a 21) to obtain a p21 1 (modulo p) B Remark: Is there a converse to Fermat s Theorem For example, suppose a n21 1 (modulo n) for every integer a with 1 gcd{a, n}.

You can add type to a document by simply clicking on the canvas and typing. You can edit text that has been added by either clicking on the text box with the Text tool or double-clicking it with the Pointer tool. You can change attributes of the text, such as the font face, for an entire block of text by selecting it with the Pointer tool and making the necessary adjustments on the Properties panel.

Does it follow that n is a prime The answer is no; for example, 561 3 11 17, and although it is not obvious, a 560 1 (modulo 561) Moreover, there are in nite numbers of such Carmichael numbers We return to the false converse of Fermat s Theorem later in this chapter when we examine the testing of numbers to determine if they are prime The Euler totient function f(n) of an integer n was de ned in Section 112 as the number of intergers less Q than n that are relative prime to n Proposition 116 gave the formula f(n) k pini 1 ( pi 1) when the prime factorization of n is i 1 n pn1pn2 pnk.

box. The Text Editor can then be opened by either selecting an existing block of text, choosing the Text tool and clicking the canvas to create a new text block, and then choosing Text Editor. Alternatively, if you already have a text block in your document, you can select it and then choose Text Editor.

26. Then, on the actions layer, add a stop(); action to frame 1. (This isn t really necessary, because Movie Clips play, once loaded, without relation to the Main Timeline and we only have one frame in our Main Timeline anyway; but for the sake of consistency, I always maintain an actions layer.)

Here we need only the special case where n is the product of 1 2 k two (distinct) primes n p1p2; in this case, f(n) ( p1 2 1)( p2 2 1) We need an important generalization of Fermat s Little Theorem Proposition 123: (Euler s Theorem): If integers n and m are relatively prime, meaning their greatest common divisor is 1, 1 gcd{m, n}, then m f(n) 1 (modulo n)..

2. Choose Commands Creative Fade Image. The Fade Image dialog box is displayed. 3. Select the type of fade desired from the eight possibilities. As you move your mouse over each of the fade styles, a red outline shows the basic shape of the fade. A white outline notes a selected style. 4. Click OK when you re done. Once the fade is applied, adjust the angle and the degree of transparency by dragging the gradient fill handles of the mask.

Figure 5-2 shows the result, which is, if I do say so myself, quite attractive and similar to the original design. By now, I hope that you can read the preceding HTML snippet and understand all the paired formatting tags. If not, you might want to skip back to s 3 and 4 and study it a bit more to refresh your memory on text-style formatting.

When copying a record that contains repeating fields, ASCII 29 (a nonprinting character known as GS, for Group Separator) separates the repetitions.

Caution You must include the letters px, for pixels, after your clip value, without a space between (for example, 20px). If you enter just the number, the clipping may not be displayed properly in either Netscape Navigator or Internet Explorer.

Pr a posteriori {Y (i) , Y (j) =ADJ(i, j)} Pr a posteriori {Y (i) , Y (j) =ADJ(i, j)}