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Chapter 1: Q26E (page 49)

What is the least significant decimal digit of (1717)17? (Hint: For distinct primesp,q, and any a is not equal to role="math" localid="1658726105638" a鈮�0(modpq), we proved the formula role="math" localid="1658726171933" a(p-1)鈮�1(modpq)in Section 1.4.2.)

Short Answer

Expert verified

The least significant decimal digit can be categorized as the rightmost digit of the decimal number written in scientific notation having mod of 10 .

Step by step solution

01

Calculate (1717)17 mod 1

We have to find least significant digit of (1717)17.

For that we need to find out modulo of (1717)17mod10.

Factors of 10 are,1,2,5,10 .

Prime number between the factor of 10 are 2 and 5.10=2,5

Formula we need to use is,

ap-1q-1=1modpq

02

Calculations

Given that,

a=17,p=2 , and q=5.

By putting these value in the formula, we get

172-15-1=1mod10174=mod10

1717can also be written as

1717=42+117

So, 171717mod10can be written as,

174脳cmod10脳17mod10=7

Where, C is constant.

The least significant decimal digit of 171717 is 7.

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Most popular questions from this chapter

Determine necessary and sufficient conditions on xandc so that the following holds: for anya,b, if ax鈮�bxmodc, thena鈮�bmodc .

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).

How about 37? To see if the number is divisible by 37, subdivide it into triples from the end(487,437,562) add these up, and see if the sum is divisible by37 .


This is true for any prime pother than2 and 5. That is, for any prime 辫鈮�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.

Show that any binary integer is at most four times as long as the corresponding decimal integer. For very large numbers, what is the ratio of these two lengths, approximately?

Wilson's theorem says that a numberis prime if and only if
(N-1)!=-1(modN).

(a) If is prime, then we know every number1鈮�x<p is invertible modulo . Which of thesenumbers is their own inverse?
(b) By pairing up multiplicative inverses, show thatrole="math" localid="1658725109805" (p-1)!=-1(modp) for prime p.
(c) Show that if N is not prime, then(N-1)!鈮�(modN) .(Hint: Considerd=gcd(N,(N-1)!.)
(d) Unlike Fermat's Little Theorem, Wilson's theorem is an if-and-only-if condition for primality. Why can't we immediately base a primality test on this rule?

If p is prime, how many elements of{0,1,...pn-1} have an inverse modulopn ?
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