Homework Review

Notes 1, Problem 1

Let a and b be positive integers, and writea

b= m.d1d2 . . . dkdk+1dk+2 . . . dk+r

where m is a positive integer, the dj are digits, and r ischosen as small as possible. Prove that r divides (b)where is Eulers function.

m = pe11 p

e22 perr

(m) =

m = pe11 p

e22 perr

(m) = pe111 per1r

m = pe11 p

e22 perr

(m) = pe111 per1r (p1 1) (pr 1)

m = pe11 p

e22 perr

(m) = pe111 per1r (p1 1) (pr 1)

Remark: If b|b, then (b)|(b).

ab= m.d1d2 . . . dkdk+1dk+2 . . . dk+r

ab= m.d1d2 . . . dkdk+1dk+2 . . . dk+r

WLOG, k = 0. Why?

ab= m.d1d2 . . . dkdk+1dk+2 . . . dk+r

WLOG, k = 0. Why?

Otherwise consider10ka

b.

ab= m.d1d2 . . . dr

ab= m.d1d2 . . . dr

(10r 1) a

b= md1d2 . . . dr m

ab= m.d1d2 . . . dr

(10r 1) a

b= md1d2 . . . dr m

(10r 1 ) a

b Z

(10r 1 ) a

b Z

(10r 1 ) a

b Z

Supposea

b=a

bwhere gcd(a, b) = 1.

(10r 1 ) a

b Z

Supposea

b=a

bwhere gcd(a, b) = 1.

(10r 1 ) a

b Z

Supposea

b=a

bwhere gcd(a, b) = 1. Then

b|

(10r 1 ) a

b Z

Supposea

b=a

bwhere gcd(a, b) = 1. Then

b|( 10r 1 ).

(10r 1 ) a

b Z

Supposea

b=a

bwhere gcd(a, b) = 1. Then

b|( 10r 1 ).IDEA: r is the order of 10 modulo b

(10r 1 ) a

b Z

Supposea

b=a

bwhere gcd(a, b) = 1. Then

b|( 10r 1 ).IDEA: r is the order of 10 modulo b

= r|(b)

(10r 1 ) a

b Z

Supposea

b=a

bwhere gcd(a, b) = 1. Then

b|( 10r 1 ).IDEA: r is the order of 10 modulo b

= r|(b)= r|(b)

(10r 1 ) a

b Z

Supposea

b=a

bwhere gcd(a, b) = 1. Then

b|( 10r 1 ).IDEA: r is the order of 10 modulo b

= r|(b)= r|(b)

Whats wrong with this IDEA?

Finish the proof by showing that if t is such thatb|( 10t 1 ),

thena

b= m.d1d2 . . . dt.

Finish the proof by showing that if t is such thatb|( 10t 1 ),

thena

b= m.d1d2 . . . dt.

So t r.

Notes 1, Problem 2a

b= m.d1d2 . . . dkdk+1dk+2 . . . dk+r

Notes 1, Problem 2a

b= m.d1d2 . . . dkdk+1dk+2 . . . dk+r

r = b 1 =

Notes 1, Problem 2a

b= m.d1d2 . . . dkdk+1dk+2 . . . dk+r

r = b 1 = (b 1)|(b) =

Notes 1, Problem 2a

b= m.d1d2 . . . dkdk+1dk+2 . . . dk+r

r = b 1 = (b 1)|(b) = b is prime

Notes 1, Problem 2a

b= m.d1d2 . . . dkdk+1dk+2 . . . dk+r

r = b 1 = (b 1)|(b) = b is prime

b 6= 2

Notes 1, Problem 2a

b= m.d1d2 . . . dkdk+1dk+2 . . . dk+r

r = b 1 = (b 1)|(b) = b is prime

b 6= 2 and b 6= 5

Notes 1, Problem 2a

b= m.d1d2 . . . dkdk+1dk+2 . . . dk+r

r = b 1 = (b 1)|(b) = b is prime

b 6= 2 and b 6= 5

gcd(a, b) = 1

Notes 1, Problem 2a

b= m.d1d2 . . . dkdk+1dk+2 . . . dk+r

r = b 1 = (b 1)|(b) = b is prime(10b1 1) a

b Z

Notes 1, Problem 2a

b= m.d1d2 . . . dkdk+1dk+2 . . . dk+r

r = b 1 = (b 1)|(b) = b is prime(10b1 1) a

b Z

= k = 0

Notes 1, Problem 2

Examples:

Notes 1, Problem 2

Examples:

1

11= 0.090909 . . .

Notes 1, Problem 2

Examples:

1

11= 0.090909 . . .

1

61= 0.01639344262295081967213114754098

3606557377049180327868852459 . . .

Notes 1, Problem 2

Examples:

1

11= 0.090909 . . .

1

61= 0.01639344262295081967213114754098

3606557377049180327868852459 . . .

Notes 1, Problem 2

Examples:

1

11= 0.090909 . . .

1

61= 0.01639344262295081967213114754098

3606557377049180327868852459 . . .

Notes 1, Problem 2

Examples:

1

11= 0.090909 . . .

1

61= 0.01639344262295081967213114754098

3606557377049180327868852459 . . .

Notes 1, Problem 2

Examples:

1

11= 0.090909 . . .

1

61= 0.01639344262295081967213114754098

3606557377049180327868852459 . . .

Notes 1, Problem 2

Examples:

1

11= 0.090909 . . .

1

61= 0.01639344262295081967213114754098

3606557377049180327868852459 . . .

Notes 1, Problem 2

Examples:

1

11= 0.090909 . . .

1

61= 0.01639344262295081967213114754098

3606557377049180327868852459 . . .

Notes 1, Problem 2

Examples:

1

11= 0.090909 . . .

1

61= 0.01639344262295081967213114754098

3606557377049180327868852459 . . .

Notes 1, Problem 2

Examples:

1

11= 0.090909 . . .

1

61= 0.01639344262295081967213114754098

3606557377049180327868852459 . . .

Notes 1, Problem 2

Examples:

1

11= 0.090909 . . .

1

61= 0.01639344262295081967213114754098

3606557377049180327868852459 . . .

Notes 1, Problem 2

Examples:

1

11= 0.090909 . . .

1

61= 0.01639344262295081967213114754098

3606557377049180327868852459 . . .

Notes 1, Problem 2

Suppose r = b 1.(i) Prove that each of the digits 0, 1, . . . , 9 occurs among

the digits d1, d2, . . . , dr either[(b 1)/10] or [(b 1)/10] + 1

times.(ii) Prove that 0 occurs [(b1)/10] times among the digits

d1, d2, . . . , dr.

b = p, a prime

b = p, a prime

Note: r = p1 = the order of 10 modulo p is p 1

b = p, a prime

Note: r = p1 = the order of 10 modulo p is p 1The values of 10ja modulo p are distinct for 0 j p 2

b = p, a prime

Note: r = p1 = the order of 10 modulo p is p 1The values of 10ja modulo p are distinct for 0 j p2 and are congruent to 1, 2, . . . , p1 in some order.

b = p, a prime

Note: r = p1 = the order of 10 modulo p is p 1The values of 10ja modulo p are distinct for 0 j p2 and are congruent to 1, 2, . . . , p1 in some order.

10ja = pqj + rj

b = p, a prime

Note: r = p1 = the order of 10 modulo p is p 1The values of 10ja modulo p are distinct for 0 j p2 and are congruent to 1, 2, . . . , p1 in some order.

10ja = pqj + rj =10ja

p= qj +

rj

p

b = p, a prime

Note: r = p1 = the order of 10 modulo p is p 1The values of 10ja modulo p are distinct for 0 j p2 and are congruent to 1, 2, . . . , p1 in some order.

10ja = pqj + rj =10ja

p= qj +

rj

p

10ja

p= d1d2 . . . dj .dj+1 . . . drd1d2 . . .

b = p, a prime

Note: r = p1 = the order of 10 modulo p is p 1The values of 10ja modulo p are distinct for 0 j p2 and are congruent to 1, 2, . . . , p1 in some order.

10ja = pqj + rj =10ja

p= qj +

rj

p

10ja

p= d1d2 . . . dj

qj

.dj+1 . . . drd1d2 . . .

b = p, a prime

Note: r = p1 = the order of 10 modulo p is p 1The values of 10ja modulo p are distinct for 0 j p2 and are congruent to 1, 2, . . . , p1 in some order.

10ja = pqj + rj =10ja

p= qj +

rj

p

10ja

p= d1d2 . . . dj

qj

.dj+1 . . . drd1d2 . . . rj/p

10ja

p= d1d2 . . . dj

qj

.dj+1 . . . drd1d2 . . . rj/p

10ja

p= d1d2 . . . dj

qj

.dj+1 . . . drd1d2 . . . rj/p

rj varies from 1 to p 1

10ja

p= d1d2 . . . dj

qj

.dj+1 . . . drd1d2 . . . rj/p

0 rjp