Number Theory 0001
Number Theory 0001
Number Theory 0001
Prime and
ussol 1 Composite
Numbers
6t lesson Outcomes
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Samuel wrles the following numbers on a piece ot paper: 48,57,37,91 and
76. He lhen asks Oave to denliry rhe number which does not belong io the group
Dave qives lhe @neci answe. What is hts answer?
@ [et! Focus
What do you lhink is the basis of Oave in identify ng the numtilr whtch is
d tierpnr tom i-e resr? I el s co-side. sol e poss,bte wdys.
1. All five numbeG nave two digits so il cannot be used as basis.
2. Two oi lhe numbers are even white three are odd, so agatn, this ctassi,icalio;
does not make any one number dttferenr.
How aborl tind ng all the faciors oi each number?
1. 2 is prime since it has only two faclors_ 1 and 2. ln fact, 2 is the small€sl
p me number.
2. 97 is prime. lt is th€ biggest two_digit pnme number'
3. 138 is obviously composite since it is €ven. Each of ih€ even numb€rs grestar
than 2 has 2 as factor olher than'1 and itselfwhich makes alleven numbers
greaisr than 2 as prime.
51 rs composrle wiih factoE of 1. 3. 17. and 51.
5. T is NEITHER pime nor composite because it has 1 lactor onlv.
24
ln our given number, lhe pime factoE arc 2 and 3 which are bolh primss.
i
product of 2 2 x 2 r 3 is 24. Therefore, our prime factorizalion of 24 is correci.
90
10
2/-\ 9 choose any pair of factors.of go We mav also choose
Z'\ Z\ from2&a53&30,5&18,and6&15'
2533
The prime fadonzaion of 90 is 2 x 3'? t 5.
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. APina Nu,,betisawhole numbergrcate.than I which hasonlytwofacioGr
and lhe number itself.
. A Composite Nunberis a whole number with three or more factors.
. Pnme Factorizalbn is a proc€ss of expressing a composiie number as a
product of its prime factors-
t=q-
t-ZL"*_Sdre
A. Write eBIME if so, and CO[,lPoSlTE if olheMise.
1. 124 -_ 6. 73 -_
2-77 -_ Z. gg ._
3. 69 -_ 8. 61 *-
4. 121 4_ 9. 43
-_
5.59+ 10. 101 +
3.123=
5. 258 =
ie ii:e *i
* E.= €E oltheo'vennumber:n.i I
$€ rhe naiu,alor6!nhn. I
.umb€6 i
^,0 ttEE *S Si
ra\ tL= st ,r , numbsr which are, fierefore,
\
.,8
Ieochlig Mdh.mot6 li ln. tnidnEdtqh G.d.;
Prime numbers between I and 100 with the Sieve of Eratosthenes
1 Piace lhe numberc from 1 io 100 in a iable like this
1 2 3 6 8 I 10
11 12 13 14 15 16 17 18 '19 20
21 22 23 24 25 26 27 2A 29 30
31 32 33 34 35 36 38 39 40
4'1 42 45 46 47 48 49 50
51 52 53 54 56 58 59 60
6l 62 63 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
8l a2 84 85 86 a7 88 89 90
91 92 93 94 95 c6 97 98 99 100
@.onih.
Y.- Rrror*,
https/www.khanacademy.olg/maih/pre-a]gebrc/pre-algebra{acrors-muttiples/
pre-algebla pime-numbers/v/.ecognizing-prime-,rumbels
https://www.youtube.com/'/vatch?v=3h4UK62Qrbo
https://www.maihgood es.com/tessons/vol3/p me composite
htlps://M.malhsisf un.com/pime-composite-numberhlmt
hllps://www.youtube.com/waich?v=Rc_2IttcOPS
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\r/- l^esoa Outcomes
At the end oflhe lesson, you should b€ able lo:
. identify the divisibility rules for on+digil numbers and sslect€d two-digit
numbels (M5NS-lb-58.1); and
. solve roulineand non,routane problems involving divisibitity rules
(ll45NS-1c,59).
rttFocus \ \
To.answerlhe Lesson Opener, we need lo know l-::-:------:\
ihe divisib,tity rures for I i,o ',0. , ."un" ,n"r. ," n""o-' i lE.|6t mC i
ro know if when rs a number divisDre or c€n be exaclty I U-- !
divided by 1.2.3. .... 10. I orru.oe,
'.isaivisiore I
For nsrance, we can say that 24 is divisibre by 8 : lll!'l"llll"jlr; ' I
since 24 + I = 3. while 24 rs not dMsible by 7 i "r,o're nr.oe, a"a y* i
because 24 : 7 = 3 wilh a ramainder of s
l_:_________.i
A Te3t for Oivisibillg by I
Every numbei is divisible by 1.
Let , be a nalural number Then 4 divides /? if, and onty if, 4 dtvides lhe
number named by the lasl lwo digiis of r. Similarty, I divides , if, and onty if, 8
divides the number named bythe lasl three diqits ofr. ln generat, 2, divides , il
and only il2,divides lhe number named by the tast /digits of r_
Let's Ch.ck:
1 . 231 is NOT divisible by 2 because il is ODD_ When divided by 2, ii wil g ve
A naiural number is divisible by 3 if, and onty if, the sum ot irs digits is
divisible by 3. Similady, a naluralnumber is divisibl€ by 9 if, and only if, the sum of
ils digils is divisible by 9.
Lel's Check:
To find out, we add 4 + 2 + 3i the answer is 9. Sinc€ I is divisible by 3, ihen
423 is divisibl€ by 3. Similady, 423 is also divisible by 9 sinc€, obviously, the sum,
which is 9, is divlsible by 9.
Lei , be a nalural number. Thenr) is dlvisible by 5 if, and only ii, its
digit is 0 or 5. 'rnit
L.t's Check:
1. 216 is divisible by 0 since it is divisible by boih 2 and 3 (faclors of 6).
2. 504 is divisibe by 12 because it is divisible by 3 and 4, two faclors of
12 which are rclalively prime.
3. I 320 is divisible by 15 since I is divisible by 3 and 5 (faciors of 15).
Let's Check:
ln 41 019, lhe digits in the odd positiofl (firsl, thnd, and itth) are 4, 0, and I and
th€ir sum is 13. on the olher hand, the digits in ihe even position (second and fou,tt)
are 1 and 1 and their sum ls 2. The difiercnce between ihe sums of lhe odd_placed and
even-placed diqits, 13 2 = 11, is divisible by 11.Ihercfore, 41 0'19 is divisible by 11.
{t rp."*.
To practice more on divisibility rules, we
r€ going to play lhe game The Boat is Sinking . Th€ t6acher will sta.l by
t$r A Twisl. The lwisl is, those siud€nts who do sayinq: '"the boat is sinking,
g@up youBelves by 6.'
,d belong to a group will be disqualifed to ioin
The students who do not
n t,le nexl rcund. While ptaying, recod on the
belong wilh any group will be
&ad the numbeE being used in t're game to
b discussed afteMards. A sample of the lable is
40 6 6
2 40-4 = 36
3
2 3 4 5 6 8 9 't0 11 16
1.234
2.450
3. 675
4.1680
5. 3740
1. 5296 i 3
2.1275:9
3.856;6
_
4. 10052 | 2
5. 976;4 _
C. Solve th€ followins probl€ms.
1- 156 pupils will go to a f6ld lrip and severalte6chels are assignod to a group
of equal numbsr of pupils. The head tsacher lhought of grouping th6 pupils in
groups of 3s, 4s, 5s, 6s, 7s, 8s and 9s. Ar€ all groupings possible? Show your
2- Find alllhe possibl€ values ofA and B if the 6-digit numb€r 2A1686 is divisible
by 4 and 9.
3. Find the smallest 4{igit numb€r which is €xactly divisible by all lh6 numbers
fmm 2 to 10.
4, lf a number is divisible by 3, 5 and 11, what as the noxt larger number diu&
bi sll t\ese numbers?
@,amrn*-""
hft ps://www.mathsisf un.com/divisibility-rules.html
https://www.khanacademy.org/math/pre-algebra/pre-algebra-factors-multiples/pre-
algebra-divisibility{ests/v/divisibility-testsjor-2-34-5-6-9- 1 0
https:i/www.mathwarehouse.com/arithmetic/numbers/divisibility-rules-and-tests.php
https://www.youtube.com/watch?v=Yl pAKJ4rf:M
https://w.youtube.com/watch?v=i 1 6N01 ldlhk
6l lessoa Outcomes
@ Lat! Focus
\-
One way to answer the prcblem above is !o simply lisl down all the possibls
whole meter pieces which each sting can be cut inio exactly. For insiance, the 48 m
stdng can be cut into I m, 2 m, 3 m, 4 m, 6 m, 8 m, 12 m, 16 m, 24 m, and 48 n
pieces. The second string, 80 m, can be cut into 1 m, 2 m, 4 m, 5 m, 8 m, 10 m, 16 m,
20 m, 40 m, and 80 m pieces. Finally, lhe 96 m stdng can be cut into 1 m, 2 m, 3 m, 4
m, 6 m, I m, 12 m, 16 m, 24 m, 32 m, 48 n, and 96 m piec€s. The ljsl is ananged in
4A - 1, 2, 3, 4, 6, 8, 12,'16, 24, 4A
To flnd the GCF of 48, 80 and 96 using the prime factorization method, simply
find the prime factorization of the given numbers-the product of the prime factors
common to all given numbers is their GCF. That is,
48= x3
80= x.
96= xix2
Com,non Prim. Factors: 2 x 2 x 2 x 2 = 16
Therefore, the GGF of 48, 80 and 96 is 16.
. 2 | 24 40 48 . Divide by 2.
2 | 12 20 24 Divids by 2.
2 | 6 10 12 Divide by 2.
3 56
Since 3, 5, 6 are already relatively prime, then the GCF of 48, 80 and 96 is the
product ofthe prime faclors used as divisors which is 2 x 2x2x2ot 16.
Chopter2:NUT.IBERtHEOtY terlon3iGr{letlCo.monfocU,teCf) 33
|
ilBlblbq
|
I
. Two or moE numbers ae said lo be retattv.ty
they have no more common diviso.s ex@pt 1.
pdme
i
f i
I
$t rpr"*L€l's lind out if which of the lhree methods in finding the cCF you tike best.
Solve for the GCF of the given sel of numbers. The ctass witt be divided in lhree
groups. Please refer 10 the table below on the assigned method for each grolp to use.
Group
1 2 3
'1.28 and 70 Lisling
2. 39. 52 and 65 Lisling
3. 36 and 54 Listing
UYleLlgELA
o Factors are nun'beE beng multiplied to gel lhe producr.
o Great*t Common Factor (cCF) is the biggest number that c€n exacity divide
lhe given numbers.
. The methods of rnding lhe GCF of lwo or morc numbers are Lisling, Pdme
Faclorization, Continuous Division, and Euclidian Atgortthm_
E\^
l-ZLet\
t&a- Solve
A. Find lhe GCF of the following numbers using the merhod indicaled.
1. 12, 14,20 Lisling 4.21, 35, 84 - Prime Faclo zalion
2. 24 and 30 - Prime Facto zalion 5. 42, 72, 90 - Conlinuols Division
3. 24, 32, 56 Continuous Division
114
2.27
3. 56
472
5.2'1
C. Solve the following problems
'1. The principal of a school wishes to diskibute 84 balls and 108 bats equally
among a number of boys. Find the groatest number of boys who will receive
the gift in this way.
2. .When a certain number of children share 208 or 125 comic books, there are
leflovers of 8 and 5 comic books, respectively. Find the largest possible number
. of children.
3. I am a single-digit number. lf I divide 39, 85, and 113, there will be remainders
of 3, 4, and 5, respectively. What is the greatest number I could possibly be?
4. Aaron wishes to group 56 oranges, 196 apples, 84 mangoes, and 140 pears
such that each group must be of the same fruit and has the same number of
fruits. What is the smallest number of groups that Aaron can make out of these
fruits?
5. Find the greatest number such that if 52 and 67 are to be divided by this
number, you get the same remainder in each case. What is the product of the
digits of this number?
^.
Ii( tetb Reflea
After leaming this lesson, reflect on how it can be learned by your future pupils in
a more effective, creative, and meaningful way. Discuss your:
1. Motivational Activlty. I will use
@q@sss,,*I
httpsr/www.mathsisfun.com/greatesl-common-faciorhtml
https/www.youlube.com/walch?v=uE908N5JYB4
https://www.youlube.com/watch?v=Nikjbvb3zv8
https:/l,!ww.puelemath.com/modules/lcm gclhtm
https:/,h/w;.khanacademy.org/math/pre-algebla/pre-algebrajaclors-multiples/pre'
algebra-greateslcommon-divisor/v/greatoslcommon-div sor
u$.r4 I
Least Common
Multiple (LCM)
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Vle$e!-qPercr
$,Vsu-Himamaylan Cily Campus has ihree bells. BellArings every 60 minules, Bell
B every 90 minutes, and BellC every 45 minules. They all ring together at 7:00 a.m. When
is rhe next rime that they will all ring together again?
The most logical way lo solve the problem is by lisiing lhe time from 7:00 and
adding successively 60 minutes or t hourforBellA,90 minutesfor&118. and45 minutes
lor Bell C until lhe first common tims emerg€s. This, however, mighl lake loo lotrg to
do. The best oplion is to solve by finding lhe Least Common Multiple ot the numbers (in
minules) and converl them to hours, then add lo 7:00.
Let us frst deline "inulliple'. Whal is multiple orwhat are mulliples ofa number.)
Multipl€s are producls of ihe natural numbeB and ihe givsn number. For inslance,
he multiples of8 are 8, 16, 24,32,40, and so on. These sre derived by mullaplying 8 by
1,2,3,4,5, and so on.
Let us now solve lhe probl€m above using lhe Lislins [4ethod.
Lblins M€thod
Step 1: List the mulliples of each number
60 - 60, 120,.180.240,300, ...
90 -90, 180, 270, 360,...
45 - 145,90, 135 180,
Since I 80 is lhe first multiple cornmon to all lhree numbers, lhen il is the
LC[,{ of lhe numbers.
60-2x2x3t5
90-2r 3r5r3
45- 3x5x3
Step 2: Mulliply lhe common multiples.
2r2x315,3=180
ObseNe th€t unlike in GCF where a pime tuctor has to be common to all given
numbers, for LCM, even if a prime factor is common to only llvo numbers, it can slill be
consid€rod as a common prime factor
Stap 2: Since ther€ is no more common prime factor for all thre€ numbeE, lhen
nnd a common pdme factor ror any two numbers. Ering down the number that is
not divisible by lhe prime divisor
3 110 15 15 Divideby3.
511055 Dvidebys.
211
9tstD,-u$.
L6t's tind oul if which of the three melhods in linding lhe LCM do you like b€st.
Solvs for lhe LCM of the'given set of numbers. The class will be divided in lhree groups.
Please reier to the lable below on the assigned melhod for each group to use.
6roup
1 2 3
3. 36 and 54
Conlinuous
Listing
Division
&fgilarq
a ,{u][plea are products of a given nomber and lhe naturarcounting numbers.
a L68t common Multiplo (Lcti,l) is t\s leasl numberlhal c€n exactly be divided
by lhe given numb€rs.
. Th8 thr€€ mothods of fnding the LCM of 2 or more numbors ar€: Lldlng,
Primo Factorlzatlon and Conllnuou3 Dlvlslon.
Solve
A. Discovering pattems
Ta6k List lhe frsl 5 common multipl€s of6,8 and 10 using the Lisling Melhod.
6-
8-
't0-
GuldeQu$tiom:
Question 2: Whal generalizalion can you make oui of lhe pattem you have
dismvercd?
1- 4A,64
2- 24,36,42
3-2,4,6,A,10,
2. A lighlhouse flash€s ils light every I 2 minules. Another lighlhouse flashes every
18 minutes. lf the two lighthouses flash logether at 12:00 noon, at what time will
they nexl fash togethe,
3. When the box of biscuits is shared equally among 12, 15, or 18 children, there
are always 8 biscuits left. Find the smallesl number of biscuits in the box.
4. I have 3 numbers. They are consecutive multiples of 3. lheir sum is 27. What is
the LcM of these 3 numb€rs
5. Find lhe least number which when divided by 12, 15, 18, and 30 gives the
remainder6.9. 12, and 24, respectivelv.
ffir*r,*"
How do you solve problems like this?
Two numbers have a GCF of 16 and an LCM of 96. lf one of the numbers is 48,
what is the other?
Let us solve this problem by doing the following activity.
1. Think of any two numbers.
2. Find their product.
3. Find their GCF and LCM..
4. Find the product of the GCF and LCM.
What can you say about the product of the given numbers and the product of
their GCF and LCM?
Product Product
Given Numbers GCF LCM
(of given numbers) lcCF x lCMl
1.
@urn*-"" https://www.khanacademy.org/math/pre-algebra/p!'e-algebra-factors-multiples/
pre-algebra-lcm/v/leasl-common-multiple-exercise
1 U3L3GL.html
http://www. math.com/school/subject'l /lessons/S
https://www.mathsisf un.com/least-common-multiple. html
https://www.youtube.com/watch?v=MjbjDvY-Kc
https://w.youtube.com/watch?v=z5vlj06Ex0U
1. 24isalac(otofl2O.'
2. 12 is a multiple of 24.
3. lfa number€nds in 0, then it is always divisible by 1,2, and
5.
4. The sum oftwo cons;utive odd numbers is always divisible
The GCF of two numbers is 8 v,tile lheir LCM as 96. lf on€ numb€r is 32, whal
is the olh€A
A. 18 C. 2A
B. 24 D. 32
Novemelcan anange his stamps by 16, 12, and 18 on a page without any left
over What is fne smallsst number of stamps for which h€ cqn do lhis?
c. 164
B. 150 D. 180
4
Di( Let's Reflea
Ater leaming this lesson, reflect on how it can be learned by your future pupils
in a more effective, creative, and meaningful way. Discuss your:
f . iiotivational Activity. I will use
"-4e9
r. z.s
-- !2 . u+
hths:/ flw.youtube.mm^flatch?v=GpumUOiGS6Q
htlps:/Ailw.youtube.com^mtch?v=TrutPJf9cmQ'
htlps:/ ilw.ksleaming.@m/ftee-math-worksheets/fifth-grade-s/converting-fEctions/improper-
fractions-tGmixed-numbe6
https:/ ilw.greatsdhools.org/gldworksheets/changing-improper-fraclions-to-mixed-numbers/
https:/ ilw.teacheBpayteachere.com/PrcducuFREE-Valentines-Day-Math-Ceoter-10661'16
Chqptq 3: le$on l: Chonglng lmEopd trqcllon lo Mlxed Number ond Vlce Vetrq
'RACIIONS