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Chapter 1: Q10E (page 48)

Show that if a鈮�b(modN)and if M Divides Nthena鈮�b(modM)

Short Answer

Expert verified

It is proved that if a=b(modN)andM dividesN then a=b(modM).

Step by step solution

01

Introduction

For,

a=b(modN)

AndM divides N.

To prove, a=b(modN).

02

To prove a≡b(modM) 

Assume that,

a=b(modN)

Then,

For some k鈭�Z,

By using division algorithm,

a鈥�and鈥�bcan be defined as,

a=nx+yb=nx'+y'

For0鈮�y andy'<n.

Then,

a=b+nk=(nx'+y')+nk=n(x'+k)+y

Comparing the above equations,

a=nx+y鈥夆赌夆赌夆赌夆赌夆赌�0鈮�y<na=n(x'+k)+y鈥夆赌�0鈮�y'<n

After division, a鈥�and鈥�bleaves the same remainder when it is divided by nthen,

a=nx+yb=nx'+y'

Subtracting these two equations,

a鈭�b=n(x鈭�x')

Thus,

a=b(modn)

Therefore, ifa=b(modN) andM divides Nthen a=b(modM).

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Most popular questions from this chapter

Give an efficient algorithm to compute the least common multiple of two n-bit numbers x and y, that is, the smallest number divisible by bothx and y. What is the running time of your algorithm as a function of n?

On page 38, we claimed that since about a 1nfraction of n-bit numbers are prime, on average it is sufficient to draw O(n)random n -bit numbers before hitting a prime. We now justify this rigorously. Suppose a particular coin has a probability p of coming up heads. How many times must you toss it, on average, before it comes up heads? (Hint: Method 1: start by showing that the correct expression is鈭�i=1鈭�i(1-p)i-1p . Method 2: if E is the average number of coin tosses, show that E=1+(1-p)E).

Show that log(n!)=胃(nlogn)

(Hint: To show an upper bound, compare (n!) with nn. To show a lower bound, compare it with (n2)n2).

Starting from the definition of x鈮�ymodN(namely, that Ndivides x-y), justify the substitution rule x鈮�x'modN,y鈮�y'modNx+y鈮�x'+y'modN,and also the corresponding rule for multiplication.

Is the difference of 530,000and6123,456a multiple of31 ?
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