5

36 Let pk be the largest power of the prime p that divides m") , when m and are nonnegative integers: Prove that k is the number of carries that occur when m i...

Question

36 Let pk be the largest power of the prime p that divides m") , when m and are nonnegative integers: Prove that k is the number of carries that occur when m is added to n in the radix p number system_ Hint: Exercise 4.24 helps here_

36 Let pk be the largest power of the prime p that divides m") , when m and are nonnegative integers: Prove that k is the number of carries that occur when m is added to n in the radix p number system_ Hint: Exercise 4.24 helps here_



Answers

If $m, n, n^{\prime} \in \mathbb{Z}$ are such that $m$ and $n$ are relatively prime and $m \mid n n^{\prime}$, then show that $m \mid n^{\prime}$. Deduce that if $p$ is a prime (which means that $p$ is an integer $>1$ and the only positive integers that divide $p$ are 1 and $p$ ) and if $p$ divides a product of two integers, then it divides one of them. (Hint: Exercise 38.)

Okay, So this question I want to prove that it is a Carmichael number if m is it to p want as a lot of bottles. I p k where all these are our numbers and piece of Jane months one is visible by anyone's one for J. So to do this, we if we let the greatest commented wise up between be in in a t one. What? This implies that the greatest common divider between B and piece of J he's also someone or James. Then what this means is by famously with ERM we have be to p J. Once one is equivalent to one more piece of Jane's life. That's a little now, because Peter J. Minus one model is visible divisor off in minus one. What this means is that there exists a constant, um, you know, go in the hole in the images. Such sense P game last by one times by sea. Is he going to end my star one? So the standings is we can white B to the n minus. One is equal to be thio piece of Jameis by one. Always to see what is this going to be equivalent to want to see either one more piece of Jane because so since this is true for all of these James from one to Kay, we can then right by famous works of art Chinese, the Chinese remainder theory that beats the end once by one is equivalent to one more piece of one has a piece of two on Nobody touch creates Okay, because this is true for our, for one lesson to Jay listening to now we know that Peter to tell Kay easy to him. So this is just one more and and hence it's a common number, by definition.

Um so this is the Fermat's little hero. We would have a power to p minus one is equivalent to one monty in which P has to be prime and a is not divisible by Pete. So we from this we can rewritten has to pee. IHS equals two Q p plus two. So Q is a functions of peace. Human p is again ought prime number hands. From this, we can denote that que iss uneven function and this lead to having an even number Ask the result so we can write que ass and expressions of to Kate to KP in which k iss some integer combining, um, this Q and the In this equation we would then have to power to the P equals to two K pete plus two taking both sides Oh, minus one plus one. And there there we have it. So this also means that every division the viser of off the number two p is off the to p minus one is off the form two k p plus one

So we're in heaven M s a positive integer and six plus one Gulf and plus one 18 plus one are old crime. We need a proof and you close to six m plus one time is 12 plus one times 18 plus one by distribution we with by distributions and multiplication of these old brackets. Together, we would then have 1296 m que plus the 96 EMS Square plus 36 m plus one. So we now apply exercise there. Um, exercise 48 that if six. M 12 m and 18 m r division er's off n minus one so and minus one then equals two 1296 m u plus the +96 m Square plus 36. So common factor we would have who common factor that we have 18 m time. 72 mm square plus 22 m plus two a divide be if they are introduced C that b equals A and Z. So from this we can see that six, um divides a n minus one. 12 AM also divided by an who and minus one and 18 and can also be divided by and minus one, we can now apply the result of exercise. 48 apply exercise 48 result. We then have that and ISS a Carmichael number.

In this question, we are asked to prove that 60 y in Cuba minus end whenever in is non negative vintages. So we are going to do induction as always, the first day and equal to zero is obvious 60 y zero, which is true We go onto the induct keeps step assume that the statement is true for in greater than or equal to zero and we will show that the 1st 1 is also true. Now for this case, we have to look at a bit more in that detail. So first on this Lep sy, if we are given that six divide this term we can factor is into this product, right? And so since it is men up off two times tree it mean board off them have to divide this whole thing. Now pay attention to this process is to divide in in plus one. Then when we shift everything we change into in plus one the n and n plus one term is still here. So to divide the whole thing right, The same wintry interior. Why one off these two when we shifted it will still be there. So true. Divide the holding as well. So the only problem that can all occurs to us that my Oscar is that went to all tree divided first because they're gonna get pushed off the product. But luckily, we have a solution for that. It to divide the first term, then to Dubai in plus one as well. Why? Because it the same thing. Plus two, right. And this one is in minus one plus two. So we add two to the number two words duty. Why? It's the same with tree. If treaty wide the first term, then tree with divi in plus two as well, because in festivities in plus one adding tree. All right, So in either case, in either kids, no matter, no matter how to entry, divide the original problem like each term up or is it no product? In the end, we still have that board two and three divi. Really, Why the new product here and this new product is in fact, did term be one fall. The term we won for in plus one statement, right is the same fact that we just shift into in plus one. And so here we have shown that it is true and inductive. That is clear. So we have proven the statement hope this is useful. Thank you


Similar Solved Questions

5 answers
An object at the origin at time t 0 has velocity; measured in meters per second, if 0 < t < 60 if 60 < t < 150 if 150 < tSketch the velocity curve Express the objects position at t 200 as a definite integral and evaluate it using formulas from plane geometry:150 200 2 dt + 150dt + 30dt 30
An object at the origin at time t 0 has velocity; measured in meters per second, if 0 < t < 60 if 60 < t < 150 if 150 < t Sketch the velocity curve Express the objects position at t 200 as a definite integral and evaluate it using formulas from plane geometry: 150 200 2 dt + 150 dt + ...
5 answers
Question 43 Not Jet = answered Marked out of 3 p Flag question Name the peptide:HNCH-NHCHNHCHNHCH;CHzOHCH_SHCHzOHSelect one: ser-cys-phe-glyb. ser-cys-tyr-gly ala-val-leu-glnd. phe-ala-val-leue. ala-gly-phe-val
Question 43 Not Jet = answered Marked out of 3 p Flag question Name the peptide: HNCH- NHCH NHCH NHCH; CHzOH CH_SH CHz OH Select one: ser-cys-phe-gly b. ser-cys-tyr-gly ala-val-leu-gln d. phe-ala-val-leu e. ala-gly-phe-val...
5 answers
#WelOnbukT1htdrant otobo conlntaComeno_hti4LuahjuceIEHCacnalca
#Wel OnbukT 1htdrant otobo conlntaComeno_ hti4 LuahjuceIEH Cacnalca...
5 answers
Which electron configuration denotes a atom in its ground state?Select one:
Which electron configuration denotes a atom in its ground state? Select one:...
5 answers
(a)Evaluate the following integral jif xz dxdydzmarks)(b)The region U lies in the first octant and is bounded by the cylinder 1? +2?=4 and the plane J=3.Express the triple integral JJf dxdydz in terms of iterated integrals in six (6) different ways_marks)(ii)Determine the value of the integral_marks )(c) Find the triple integral by using suitable coordinate forJILb7dxdydzX2 Where the region U is bounded by a? b7=.
(a) Evaluate the following integral jif xz dxdydz marks) (b) The region U lies in the first octant and is bounded by the cylinder 1? +2?=4 and the plane J=3. Express the triple integral JJf dxdydz in terms of iterated integrals in six (6) different ways_ marks) (ii) Determine the value of the integr...
5 answers
Find an equation of the conic satisfying the given conditions.Parabola, vertex $(2,2)$, focus $left(frac{3}{2}, 2ight)$
Find an equation of the conic satisfying the given conditions. Parabola, vertex $(2,2)$, focus $left(frac{3}{2}, 2 ight)$...
5 answers
Find the indefinite integral and check the result by differentiation.$$int(1+3 t) t^{2} d t$$
Find the indefinite integral and check the result by differentiation. $$int(1+3 t) t^{2} d t$$...
5 answers
Find $mathbf{u} cdot(mathbf{v} imes mathbf{w})$$$mathbf{u}=mathbf{i}, mathbf{v}=mathbf{i}+mathbf{j}, mathbf{w}=mathbf{i}+mathbf{j}+mathbf{k}$$
Find $mathbf{u} cdot(mathbf{v} imes mathbf{w})$ $$ mathbf{u}=mathbf{i}, mathbf{v}=mathbf{i}+mathbf{j}, mathbf{w}=mathbf{i}+mathbf{j}+mathbf{k} $$...
5 answers
Explain the four paths to cooperation (chap 10)?
Explain the four paths to cooperation (chap 10)?...
1 answers
In Exercises $21-30,$ use the properties of logarithms to expand the logarithmic expression. $$ \ln \sqrt{a-1} $$
In Exercises $21-30,$ use the properties of logarithms to expand the logarithmic expression. $$ \ln \sqrt{a-1} $$...
5 answers
A projectile is launched straight up in the air. Its height (infeet) t seconds after launch is given by the functionf(t)=−16t2+355t+6. Find the time when it strikes the ground. Thetime is nothing seconds
A projectile is launched straight up in the air. Its height (in feet) t seconds after launch is given by the function f(t)=−16t2+355t+6. Find the time when it strikes the ground. The time is nothing seconds...
5 answers
Problem 3. plucked string poiuts] Consider string extendexl between rigid wlls spatraterl by distance The string initially displaced dlistanc Erom O-(quater of its length to threr-quautter of its length as shown in Figure BI At / = @ the clisplacement of the string is cescribeel bwy0 <' < 1'4(t =0,2) =<I<T +IhSt<LFigurepwints] Final the aplituck" of vach noral W '
Problem 3. plucked string poiuts] Consider string extendexl between rigid wlls spatraterl by distance The string initially displaced dlistanc Erom O-(quater of its length to threr-quautter of its length as shown in Figure BI At / = @ the clisplacement of the string is cescribeel bwy 0 <' <...
5 answers
The following data were obtained for the reaction 2C1O2 (aq) + 2OH " (aq) CIO: (aq) + C1Oz - (aq) + HzO() A[C1Oz] where Rate At [CIO2Jo [OH-Jo Initial Rate (moVL) (moVL) (moVL"s) 0.0500 0.100 5.75 * 10 7 0.100 0.100 2.30 * 10-1 0.100 0.0500 1.15 10-1Determine the rale law:(Use k for the rate constant )Rate
The following data were obtained for the reaction 2C1O2 (aq) + 2OH " (aq) CIO: (aq) + C1Oz - (aq) + HzO() A[C1Oz] where Rate At [CIO2Jo [OH-Jo Initial Rate (moVL) (moVL) (moVL"s) 0.0500 0.100 5.75 * 10 7 0.100 0.100 2.30 * 10-1 0.100 0.0500 1.15 10-1 Determine the rale law: (Use k for the ...
5 answers
Continued)Method Calculate the pH using the NH: base dissociation reaction:LQOInital Cunceztration (M}Czince Curcezlration (M) Equilibrium concentlomEquilbziun Concenlrilioz CalcualedtnShuw YOur work here Ur replicale this on & xpirate piece ol pilper_ Numbe- und Jabe- cuch cuculaton; inc.udt uniia Siuv Ihe mniln nehotneo LCL YOUT unter Repor: "nswet; unt CottuS significxt [JJur:>
Continued) Method Calculate the pH using the NH: base dissociation reaction: LQO Inital Cunceztration (M} Czince Curcezlration (M) Equilibrium concentlom Equilbziun Concenlrilioz Calcualedtn Shuw YOur work here Ur replicale this on & xpirate piece ol pilper_ Numbe- und Jabe- cuch cuculaton; inc....

-- 0.027956--