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Chapter 1: Q10E (page 48)

Show that if a≡b(modN)and if M Divides Nthena≡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∈Z,

By using division algorithm,

a and bcan be defined as,

a=nx+yb=nx'+y'

For0≤y andy'<n.

Then,

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

Comparing the above equations,

a=nx+y      0≤y<na=n(x'+k)+y  0≤y'<n

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

a=nx+yb=nx'+y'

Subtracting these two equations,

a−b=n(x−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

1.36. Square roots. In this problem, we'll see that it is easy to compute square roots modulo a prime pwith p≡3(mod4).
(a) Suppose p≡3(mod4). Show that(p+1)/4 is an integer.
(b) We say x is a square root of a modulo p if a=x2(modp). Show that if p≡3(mod4)and if a has a square root modulo p, thena(p+1)/4 is such a square root.

In the RSA cryptosystem, Alice’s public key (N,e)is available to everyone. Suppose that her private key d is compromised and becomes known to Eve. Show that e=3if (a common choice) then Eve can efficiently factor N.

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1.38. To see if a number, say 562437487, is divisible by 3, you just add up the digits of its decimalrepresentation, and see if the result is divisible by role="math" localid="1658402816137" 3.

( 5+6+2+4+3+7+4+8+7=46, so it is not divisible by 3).

To see if the same number is divisible by 11, you can do this: subdivide the number into pairs ofdigits, from the right-hand end(87,74,43,62,5) , add these numbers and see if the sum is divisible by11 (if it's too big, repeat).

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This is true for any prime pother than2 and 5. That is, for any prime p≠2,5, there is an integer rsuch that in order to see ifp divides a decimal number n, we breakn into r-tuples of decimal digits (starting from the right-hand end), add up these r-tuples, and check if the sum is divisible by p.

(a) What is the smallest rsuch for p=13? Forp=13 ?

(b) Show thatr is a divisor of p-1.
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