[PDF] Junior Math Circles February 3 2010 Number Theory I





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Problem of the Week

Determine the number of six-digit palindromic numbers which are divisible The numbers between 10 and 46 that are divisible by 3 are 1215



Junior Math Circles February 3 2010 Number Theory I

3 févr. 2010 Divisible by Test. Example. 2. The ones digit is even. 46 is divisible by 2 since 6 is even. 3. The sum of all the digits is di-.



Answers

(a) Numbers divisible by 4 (b) Numbers divisible by 3. (c) 4 8



Division Worksheet -- Divisibility Rules for 3 6 and 9 (2 Digit Numbers)

Divisible by 3? 62. 41. 26. 54. 40. 75. 98. 48. 73. 46. 23. 72. 87. 68. 23. 25. 80. 92. 88. 11. 41. 55. 32. 49. Divisible by 6?



SIMPLE DIVISIBILITY RULES FOR THE 1st 1000 PRIME NUMBERS

(3) the integer A in Eq. (1) making it possible to customize the rule. 46. A number is divisible by 199 if ? 179 times the last digit of the number ...



Fermats Little Theorem Solutions

27 sept. 2015 Thus 128129 ? 91 ? 9 mod 17. 4. (1972 AHSME 31) The number 21000 is divided by 13. What is the remainder? [Solution: 21000 ? 3 ...



THE [46 9

https://www.aimsciences.org/article/exportPdf?id=76a251a1-07ca-4310-ad63-7b484ea4efcf



Untitled

2x 46+8. 5x. 8 4 6. 1 x 6. 3+2. RES. 46. 8. 6 gcd (10046)=2 Use congruences to show that n(n + 7) is divisible by 2. is even



Math 46 Winter 2009

http://nebula2.deanza.edu/~karl/Classes/Exams/Math46.TaHo.problems.W09.pdf



UNIT 1LEARNING GUIDE –NUMERACY

Wei has prepared a playlist of 46 songs for the school dance. The dance starts at 7pm and The sum of the digits of the number is divisible by 3.



Divisibility Rules for 3 6 and 9 (A) - Math-Drills

Divisible by 3 and 9? 91 46 26 32 98 90 25 61 54 58 34 77 81 23 69 79 26 30 56 36 96 53 95 71 Math-Drills Com Divisibility Rules for 3 6 and 9 (D)



Searches related to 46 divisible par 3 PDF

Apply the divisibility rules for 2 3 and 5 If the number is divisible by 2 3 or 5 write which ever answer is true You do not need to write all of the answers (example: 110 is divisible by both 2 and 5 you may answer 2) If the number is not divisible by these rules enter Ø

Comment calculer la divisibilité par 3 ?

Cela signifie que le chiffre des unités doit être pair, c’est-à-dire 0, 2, 4, 6 ou 8 (par exemple, le chiffre des unités de 48 est 8). Exemple : 48 est une chiffre pair. Il est donc un multiple de 2. Critère de divisibilité par 3 : la somme des chiffres qui composent le nombre est un multiple de 3 (divisible par 3).

Comment savoir si un nombre entier est divisible par 3 ?

Un nombre entier est divisible par 3 si la somme de ses chiffres est un multiple de 3 (3 ; 6 ; 9 ; etc.). 534 est divisible par 3 car 5 + 3 + 4 = 12 et 12 = 4 × 3. Un nombre entier est divisible par 4 si le nombre formé par ses deux derniers chiffres est un multiple de 4. 1 028 est divisible par 4 car 28 est un multiple de 4 (28 = 4 × 7).

Quels sont les critères de divisibilité ?

Les critères de divisibilité sont des règles qui permettent de déterminer rapidement si un nombre entier (un nombre sans décimale ni fraction) est divisible par tel ou tel chiffre ou nombre. En d’autres termes, ils permettent de savoir si un nombre est le multiple d’un chiffre ou d’un autre nombre.

Junior Math Circles February 3 2010 Number Theory I 1

University of Waterloo Centre for Education in

Faculty of Mathematics Mathematics and Computing

Junior Math Circles

February 3, 2010

Number Theory I

Factors

Denition: Afactorof a number is a whole number that divides evenlyinto the other number. So, when you divide by the factor the remainder is zero. Factors come in pairs. When you divide a number by one of its factors, the an- swer or quotient is the paired factor.

Is 4 a factor of 36?

Yes, 36 divided by 4 gives us an answer of 9 with no remainder.

Is 3 a factor of 47?

No, 47 divided by 3 gives us an answer of 15 with a remainder of 2. Exercise 1State all the factors of the following numbers.

1) 50: 2) 15: 3) 13: 4) 36:

1;2;5;10;25;501 ;3;5;151 ;131 ;2;3;4;6;9;12;18;36

Notice that 36 has an odd number of factors. Each factor still has a pair, but 6 pairs with itself, and is only listed once. 2

Divisibility Rules

The rules listed in the table below help us determine whether a number is divisible

by each of the numbers from 2 to 11. Note that 0 is divisible by all positive numbers.Divisible byTestExample

2The ones digit is even.46 is divisible by 2 since 6 is even.

3The sum of all the digits is di-

visible by 3.153 is divisible by 3 since 1+5+3 =

9, which is divisible by 3.4The last two digits are divisi-

ble by 4.820 is divisible by 4 since 20 is di- visible by 4.5The ones digit is 0 or 5.795 is divisible by 5 since its last digit is 5.6The number is divisible by 2 and 3.258 is divisible by 6 since 8 is even so it is divisible by 2 and 2+5+8 =

15 which is divisible by 3.7Subtract twice the ones digit

from the rest of the number, until you get a small number that is divisible by 7.672 is divisible by 7 since 22 = 4 and 674 = 63, which is divisible by 7.8The last three digits are divis- ible by 8.4816 is divisible by 8 since 816 is divisible by 8.9The sum of all the digits is di- visible by 9.567 is divisible by 9 since 5+6+7 =

18 which is divisible by 9.10The ones digit is 0.270 is divisible by 10 since the ones

digit is 0.11The alternating sum of the digits is divisible by 11.1364 is divisible by 11 since 1-3+6-

4=0, which is divisible by 11.Exercise 2Using the rules above, check o each of the numbers from 2 to 11

that each number is divisible by.234567891011 926X

1420XXXX

7848XXXXXX

22176XXXXXXXX

39208XXX

3

Prime Numbers

Denitions:

Aprime numberis a number that has only two factors, 1 and itself. Acomposite numberit is a number with factors other than 1 and itself. Exceptional Case: The number 1 is neither prime nor composite by denition. Exercise 3Determine whether the following numbers are prime or composite. If it is composite, list out its factors.

1) 19 2) 22

prime comp osite- 1 ;2;11;22

3) 37 4) 51

prime comp osite- 1 ;3;17;51

Are there any even numbers that are prime?

Yes, 2 is the only prime number that is even. Think about why.

How can we nd prime numbers?

We can nd them using the method calledSieve of Eratosthenes.

Instructions:

1.

Cross o 1 since it is not prime.

2.

Find the rst prime n umber(2) and circle it.

3. Go through the rest of the c hartand cross o all of the m ultiplesof t wo. 4. Go bac kto the b eginningof the c hartand nd the rst n umberthat is not crossed o. Circle it, it is prime. Then go through the rest of the chart, crossing o the multiples of this number. 5. Rep eatstep 4, un tilev eryn umberin the c hartis e ithercircled or crossed o. 6. The n umbersthat are circled are the prime n umbers. 4 Exercise 4Use this method to nd all the prime numbers from 1 to 100.12345678910

11121314151617181920

21222324252627282930

31323334353637383940

41424344454647484950

51525354555657585960

61626364656667686970

71727374757677787980

81828384858687888990

919293949596979899100

Write down the primes less than 100. There should be 25 of them.

Prime Factorization

One way of describing numbers is by breaking them down into a product of their prime factors. This is called prime factorization. Every positive number can be prime factored. By denition the prime factorization of a prime number is the num- ber itself, and the prime factorization of 1 is 1. ExampleLet's prime factor 270 using a factor tree. SolutionFirst think of a factor of 270. We don't want to use 1 or 270.

Let's use 10.

Now what factor is paired with 10? That factor is 27, since 1027 = 270.270 1027
5

Is 10 prime? No, so we have to prime factor 10.

What is a pair of factors of 10? A pair is 2 and 5. Both 2 and 5 are prime, so we are done this part.270 1027

25Is 27 prime? No, so we have to prime factor it.

What is a pair of factors? One pair is 3 and 9.

The number 3 is prime, but 9 has factors 3 and 3.

270
1027
25
33

39We have decomposed 270 into all prime numbers, so now we can write it as a product

of these primes.

270 = 23335

Note: Prime factorizations are unique. Every number has only one prime factoriza- tion, and no two numbers have the same prime factorization. Exercise 5Find the prime factorization of 2520 using a factor tree. 6

Problem Set

1. Ms. Timson w antsto split her G rade9 math class of 36 studen tsin togroups for an upcoming assignment. List all the possibilities of groups, each with the same number of students, that Ms. Timson can divide her class into so that no students are left without a group. 2.

What is the rst prime n umbergreater than 200?

3. Dra wt wodieren tfactor trees for the n umber100. (Start with t wodieren t pairs of factors.) 4.

The v olumeof a cereal b oxis 1925 cm

3. What are the dierent possible dimen-

sions of the cereal box? (Note: The volume of a rectangular box islengthwidthheight.) 5. The eigh tdigit n umber1234 678 is divisible by 11. What is the digit? 6. The four digit n umber43 is divisible by 3, 4 and 5. What are the last two digits? 7. (a) If a n umberis divisible b y2 and 3, is it alw aysdivisible b y6? (b) If a n umberis divisible b y2 and 4, is it alw aysdivisible b y8? 8. What is the smallest n umberthat y oum ustm ultiply48 b yso that the pro duct is divisible by 45? 9. The pro ductof three dierentpositive integers is 144. What is the maximum possible sum of these three integers? 10. The digits 1 ;2;3;4;5 and 6 are each used once to compose a six digit number abcdef, such that the three digit numberabcis divisible by 4,bcdis divisible by 5,cdeis divisible by 3 anddefis divisible by 11. Determine all possible assignments of the digits to the letters. 11. What is the s mallestn umberthat y oum ustm ultiply1512 b yin order to get a perfect square. Aperfect squareis a number that is equal to another number multiplied by itself. For example 66 = 36 is a perfect square. 12. If xandyare two-digit positive integers withxy= 555, what isx+y? 7

Answers

1. The p ossibilitiesare: 1 group of 36 studen ts,2 groups of 18 studen ts,3 gro ups of 12 students, 4 groups of 9 students, 6 groups of 6 students, 9 groups of 4 students, 12 groups of 3 students, 18 groups of 2 students and 36 groups of 1 student. 2. 211
3.100 4 25 2 255 100
10 10

2 5524.The dieren tp ossibledimensions are:

1cm1 cm1925 cm 1cm35 cm55cm

1 cm5 cm385 cm 5 cm5 cm77 cm

1cm7 cm275 cm 5 cm7 cm55 cm

1 cm11 cm175 cm 5 cm11 cm35 cm

1 cm25 cm77 cm 7 cm11 cm25 cm

5. 9 6.

20 or 80 - The n umbercould b e43 20or 4380.

7. (a) Y es,b yde nition,if a n umberis divisible b y2 and b y3 then it is divisibl e by 6. (b) No, for example, in the exercise in the lesson ab ove,142 0is divisible b y4 and by 2 but it is not divisible by 8. 8. 15. 9.

75. The three divisors are 72, 2, and 1.

10.abcdef= 324561

11. 42

12.x= 15 andy= 37, sox+y= 52.

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