000RM.dvi

520 Repunits

18.3 Sam Yates’ repunit riddles ...........

1.What digits should be substituted for the letters so that the sum of

the nine identical addends will be a repunit?^1

REP UN IT S

REP UN IT S

REP UN IT S

REP UN IT S

REP UN IT S

REP UN IT S

REP UN IT S

REP UN IT S

+ REP UN IT S

2.Are two repunits with consecutive even numbers as their subscripts

relatively prime?^2

3.Are two repunits with consecutive numbers as their subscripts rel-

atively prime?^3

4.Are two repunits with consecutive odd numbers as their subscripts

relatively prime?^4

5.Aside from 3, what primepdividesRp?^5

6.What digit does each letter of this multiplication represent?^6

RR RRRRR

× RR RRRRR

REPUNIT INUPER

7.An old car dealer’s record in the 1960’s shows that the total receipts

for the sale of new cars in one year came to 1,111,111.00 dollars.

If each car had eight cylinders and was sold for the same price as

each other car, how many cars did he sell?

(^1) REPUNITS=12345679 and the sum isR 9.
(^2) No. Repunits with even subscripts are divisible by 11.
(^3) Yes. Consecutive numbers are relatively prime, andgcd(Rm,Rn)=Rgcd(m,n).
(^4) Yes. Consecutive odd numbers are relatively prime.
5 None.
(^6) R 7 ×R 7 =R 13.