18.5 The period length of a prime 527
Exercise
1.What are the decimal expansions of 21 k, 51 k, and 2 h^1 · 5 k?2.Calculate the period ofp=67.3.Ifpis any odd prime, show that the decimal expansion of the frac-
tion^1 pwill repeat inp− 21 digits or some factor thereof it and only if
p≡± 3 k (mod 40).4.What is the smallest integer of 2 nidentical digits which is the prod-
uct of twon-digit numbers?
Clearlyn≥ 2. Since1111 = 11× 101 , we seek 6 digit numbers.
Now,
R 6 =3× 7 × 11 × 13 × 37.
There are four ways of rearranging it as a product of two 3-digit
numbers:143 ×777 = 231×481 = 259×429 = 273× 407.5.LetN be an integer ofpdigits. If the last digit is removed and
placed before the remainingp− 1 digits, a new number ofpdigits is
formed which is^1 nth of the original number. Find the most general
such numberN.
6.(a) Find the smallest integerNsuch that, if the units digits is trans-
posed from right to left, a numberMis obtained whereM=5N.^107.A certain 3-digit number yields a quotient of 26 when divided by
the sum of its digits. If the digits are reversed, the quotient is 48.
What is the smallest 3-digit number for which this is possible?8.Without the use of tables, find the smallest integer whose cube ter-
minates in seven sevens.^1110. N = 714285 = 5M; = 142857N
11. = 901639512372747777777^39660753