Download 2011 – 2012 Log1 Contest Round 2 Theta Number Theory Name: 4

2011–2012Log1ContestRound2
ThetaNumberTheory
Name:__________________
4pointseach
1
Howmanyprimenumbersarelessthan50?
2
Twinprimesareprimeswhosedifferenceis2.Whatisthelargesttwo‐digittwin
primepair?
3
Findtheproductofthegreatestcommonfactorandtheleastcommonmultipleof14
and30?
4
Evaluate11
5
PlayersofthegameMouseareawarded0,5or9pointsoneveryturn.Somepoint
totals,suchas11arenotpossible.Whatisthelargestpointtotalthatisnotpossible?
111
1111 andexpressasabase5number.
5pointseach
6
Howmanyzeroesareattheendof111!?
7
Howmanypairsofpositiveintegers,m andn,aretheresuch 7
8
Whatistheprobabilitythatarandomlyselectedpositivefactorof5,040iseven
(expressyouranswerasareducedfraction)?
9
Whatistheleastcommonmultipleofallthefactorsof64?
10
In2011,thesumofthefirsttwodigitsequalsthesumofthelasttwodigits.How
manynumbersbetween2000and2500satisfythiscondition?
5
400?
6pointseach
11
Findtheunitsdigitof3
12
Findthesumofthepositivefactorsof720.
13
Whatisthehundredsdigitof2011 ?
14
Giventhat ≡ 3mod7,whatistheremainderwhen
15
Thedigitalrootofthenumberisthesumofitsdigitstakenuntilasumlessthan10is obtained.Forexample,thedigitalrootof399is3since3+9+9=21andthen2+1=3.
Whatisthedigitalrootof10factorial?
5
7
11 .
3
4isdividedby7?
2011–2012Log1ContestRound2
AlphaNumberTheory
Name:__________________
4pointseach
1
Howmanyprimenumbersarelessthan50?
2
Twinprimesareprimeswhosedifferenceis2.Whatisthelargesttwo‐digittwin
primepair?
3
Findtheproductofthegreatestcommonfactorandtheleastcommonmultipleof14
and30?
4
Convertthebase3number211022 tobase9.
5
PlayersofthegameMouseareawarded0,5or9pointsoneveryturn.Somepoint
totals,suchas11arenotpossible.Whatisthelargestpointtotalthatisnotpossible?
5pointseach
6
Howmanyzeroesareattheendof111!?
7
Howmanypairsofpositiveintegers,m andn,aretheresuch 7
8
Whatistheprobabilitythatarandomlyselectedpositivefactorof5,040iseven
(expressyouranswerasareducedfraction)?
9
Aperfectnumberisapositiveintegersuchthatthesumofitsproperpositivefactors
equalsitself.Whatisthesumofthesecondperfectnumberandtheseventhprime
number?
10
In2011,thesumofthefirsttwodigitsequalsthesumofthelasttwodigits.How
manynumbersbetween2000and2500satisfythiscondition?
5
400?
6pointseach
11
Findtheunitsdigitof3
12
Findthesumofthepositivefactorsof720.
13
Whatisthehundredsdigitof2011 ?
14
Howmanypositiveintegerslessthanorequalto1000arerelativelyprime(no
commonfactorsotherthan1)to100?
15
Thedigitalrootofthenumberisthesumofitsdigitstakenuntilasumlessthan10is obtained.Forexample,thedigitalrootof399is3since3+9+9=21andthen2+1=3.
Whatisthedigitalrootof10factorial?
5
7
11 .
2011–2012Log1ContestRound2
MuNumberTheory
Name:__________________
4pointseach
1
Howmanyprimenumbersarelessthan50?
2
Twinprimesareprimeswhosedifferenceis2.Whatisthelargesttwo‐digittwin
primepair?
3
Findtheproductofthegreatestcommonfactorandtheleastcommonmultipleof14
and30?
4
Convertthebase3number211022 tobase9.
5
Twodigits,rands,aresuchthatthenumber42r611s37isdivisibleby11.Whatis
theremainderwhenr+sisdividedby11?
5pointseach
6
Howmanyzeroesareattheendof111!?
7
Howmanypairsofpositiveintegers,m andn,aretheresuch 7
8
Whatistheprobabilitythatarandomlyselectedpositivefactorof5,040iseven
(expressyouranswerasareducedfraction)?
9
Aperfectnumberisapositiveintegersuchthatthesumofitsproperpositivefactors
equalsitself.Whatisthesumofthesecondperfectnumberandtheseventhprime
number?
10
Whatisthesmallestfour‐digitnumberthathasexactly3positivefactors?
5
400?
6pointseach
11
Findtheunitsdigitof3
12
Findthesumofthepositivefactorsof720.
13
Whatisthehundredsdigitof2011 ?
14
Howmanypositiveintegerslessthanorequalto1000arerelativelyprime(no
commonfactorsotherthan1)to100?
15
Whatisthesumoftheintegervalues,n,sothat
5
7
11 .
isalsoaninteger?
2011–2012Log1ContestRound2
ThetaNumberTheory
Name:__________________
4pointseach
1
Howmanyprimenumbersarelessthan50?
2
Twinprimesareprimeswhosedifferenceis2.Whatisthelargesttwo‐digittwin
primepair?
3
Findtheproductofthegreatestcommonfactorandtheleastcommonmultipleof14
and30?
4
Evaluate11
5
PlayersofthegameMouseareawarded0,5or9pointsoneveryturn.Somepoint
totals,suchas11arenotpossible.Whatisthelargestpointtotalthatisnotpossible?
111
15
71,73
420
401
1111 andexpressasabase5number.
31
5pointseach
6
Howmanyzeroesareattheendof111!?
26
7
Howmanypairsofpositiveintegers,m andn,aretheresuch 7
8
Whatistheprobabilitythatarandomlyselectedpositivefactorof5,040iseven
(expressyouranswerasareducedfraction)?
9
Whatistheleastcommonmultipleofallthefactorsof64?
10
In2011,thesumofthefirsttwodigitsequalsthesumofthelasttwodigits.How
manynumbersbetween2000and2500satisfythiscondition?
5
400?
11
4
5
64
25
6pointseach
8
11
Findtheunitsdigitof3
12
Findthesumofthepositivefactorsof720.
13
Whatisthehundredsdigitof2011 ?
14
Giventhat ≡ 3mod7,whatistheremainderwhen
15
Thedigitalrootofthenumberisthesumofitsdigitstakenuntilasumlessthan10is
obtained.Forexample,thedigitalrootof399is3since3+9+9=21andthen2+1=3.
Whatisthedigitalrootof10factorial?
5
7
11 .
2418
6
3
4isdividedby7?
0
9
2011–2012Log1ContestRound2
AlphaNumberTheory
Name:__________________
4pointseach
1
Howmanyprimenumbersarelessthan50?
15
2
Twinprimesareprimeswhosedifferenceis2.Whatisthelargesttwo‐digittwin
primepair?
3
Findtheproductofthegreatestcommonfactorandtheleastcommonmultipleof14
and30?
4
Convertthebase3number211022 tobase9.
5
PlayersofthegameMouseareawarded0,5or9pointsoneveryturn.Somepoint
totals,suchas11arenotpossible.Whatisthelargestpointtotalthatisnotpossible?
71,73
420
738
31
5pointseach
6
Howmanyzeroesareattheendof111!?
26
7
Howmanypairsofpositiveintegers,m andn,aretheresuch 7
8
Whatistheprobabilitythatarandomlyselectedpositivefactorof5,040iseven
(expressyouranswerasareducedfraction)?
9
Aperfectnumberisapositiveintegersuchthatthesumofitsproperpositivefactors
equalsitself.Whatisthesumofthesecondperfectnumberandtheseventhprime
number?
10
In2011,thesumofthefirsttwodigitsequalsthesumofthelasttwodigits.How
manynumbersbetween2000and2500satisfythiscondition?
5
400?
11
4
5
45
25
6pointseach
8
11
Findtheunitsdigitof3
12
Findthesumofthepositivefactorsof720.
13
Whatisthehundredsdigitof2011 ?
14
Howmanypositiveintegerslessthanorequalto1000arerelativelyprime(no
commonfactorsotherthan1)to100?
15
Thedigitalrootofthenumberisthesumofitsdigitstakenuntilasumlessthan10is
obtained.Forexample,thedigitalrootof399is3since3+9+9=21andthen2+1=3.
Whatisthedigitalrootof10factorial?
5
7
11 .
2418
6
400
9
2011–2012Log1ContestRound2
MuNumberTheory
Name:__________________
4pointseach
1
Howmanyprimenumbersarelessthan50?
15
2
Twinprimesareprimeswhosedifferenceis2.Whatisthelargesttwo‐digittwin
primepair?
3
Findtheproductofthegreatestcommonfactorandtheleastcommonmultipleof14
and30?
4
Convertthebase3number211022 tobase9.
5
Twodigits,rands,aresuchthatthenumber42r611s37isdivisibleby11.Whatis
theremainderwhenr+sisdividedby11?
71,73
420
738
0
5pointseach
6
Howmanyzeroesareattheendof111!?
26
7
Howmanypairsofpositiveintegers,m andn,aretheresuch 7
8
Whatistheprobabilitythatarandomlyselectedpositivefactorof5,040iseven
(expressyouranswerasareducedfraction)?
9
Aperfectnumberisapositiveintegersuchthatthesumofitsproperpositivefactors
equalsitself.Whatisthesumofthesecondperfectnumberandtheseventhprime
number?
10
Whatisthesmallestfour‐digitnumberthathasexactly3positivefactors?
5
400?
11
4
5
45
1369
6pointseach
11
Findtheunitsdigitof3
12
Findthesumofthepositivefactorsof720.
13
Whatisthehundredsdigitof2011 ?
14
Howmanypositiveintegerslessthanorequalto1000arerelativelyprime(no
commonfactorsotherthan1)to100?
15
Whatisthesumoftheintegervalues,n,sothat
5
7
11 .
8
2418
6
400
4
isalsoaninteger?
2011–2012Log1ContestRound2
NumberTheorySolutions
Solution
Mu
Al
Th
1
1
1
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47
2
2
2
Somemayput89,91but91=7*13.
3
3
3
TheproductoftheGCFandtheLCMwillbetheproductofthetwonumbers.14*30=
420.
4
4
Thebase[9]isnotneeded.Since9
intoabase9digit.
4
3
5
Divisibilityfor11meansthat(4+r+1+s+7)‐(2+6+1+3)isamultipleof11.Thismeans
thatr+sis0or11.
5
5
5x9–5–9=31
6
6
6
111/5=22r1;22/5=4r2;4/5=0r4.22+4+0=26
7
7
7
55(7)+3(5)=400,so(55,3)isonesuchpair.Onecanreducemby5andincreasenby7
withoutchangingthetotal.Thiscontinuesuntil5(7)+73(3).11total.
8
8
8
5040isdivisibleby16or24 sothat4/5ofthefactorswillbedivisiblebyapowerof2
–weexclude20=1.Thus4/5ofthemwillbeeven.Onecancountthefactorsby
completelyfactoring5040butitisnotneeded.
9
9
Thefirsttwoperfectnumbersare6and28.28+17=45
9
Thefactorsof64areallpowersof2anddivideinto64.
10
Ifanumberhas3factors,itisthesquareofaprimenumber.Thefirstprimenumber
1369.
witha4‐digitsquareis37.37
10
10
Byexhaustion.Considerthefirsttwodigits20willgowith02,20,and11(3numbers).
21willhave4numbers,22(5numbers)23(6numbers)and24(7numbers).
3+4+5+6+7=25.
11
11
11
Thepowersof3endrespectivelyin3,9,7,1,…Allpowersof5endin5,thepowersof
7in7,9,3,1andfinallyallpowersof11endin1.Thereforetheanswerendinthe
samedigitas7(5)+3(1)=38withaunitsdigitof8.
12
12
12
720 2
3
5Thesumofthepositivefactorswillbe 1
3
31 13 6
2418
3 1 5
13
13
13
2011
2000 11
2000
⋯ 11 .Allthetermsexceptthelasthaveatleast
9 8 10 1
1 .
3zeroesattheend.11
10 1
10
⋯ 9 7 10 1
Onlythelastthreetermsneedbeconsidered(theresthaveafactor1000);
3600+900+1=3691andthehundredsdigitis6.
14
14
14
15
13
85
15
15
4 25
1
2
2
2
2
1
Theprimefactorsof100are2and5.Oneneedstoeliminateallmultiplesof2and5.
1000(1/2)(4/5)=400.
≡ 9mod7and3 ≡ 9 mod 7so
willbethe0.
6, 0, 2,
101
3 ,onecanconverttwobase‐3digitsdirectly
3
4 ≡9
9
4 ≡ 0mod7,theremainder
4
sothat mustalsobeanintegerand
8.Thesesumto4.
1
1, 7,
Thedigitalrootistheremainderwhendividedby9(or9ifitisdivisibleby9).Since
10!isclearlydivisibleby9,itsdigitalrootis9.