APUNTES DEL TEMA 2
1.- MULTIPLES AND FACTORS
1.1.- Concept of multiple.
1.2.- Concept of factor.
1.3.- The properties of multiples and factors.
2.- PRIME AND COMPOSITE NUMBERS
A prime number only has two factors: the number one and itself. For example: 3, 5, 11, 17, etc. A composite number has more than two factors. For example: 4, 9, 15, 30, etc.
3.- DIVISIBILITY RULES
Las reglas de divisibilidad te ayudan a saber si un número es múltiplo de otro sin hacer la división.
- Rule of number 2: A number is divisible by 2 if its last digit is either 0 or an even number. Un número es divisible por 2 si su última cifra es 0 ó un número par. Example: 46,200, 34, 108…..
- Rule of number 3: A number is divisible by 3 if the sum of its digits is a multiple of 3. Un número es divisible por 3 si la suma de sus cifras es múltiplo de 3. Example: 45, 105, 300, 417….
- Rule of number 4: A number is divisible by 4 if its two last digits are multiples of 4. Un número es divisible por 4 si sus dos últimas cifras son múltiplo de 4. Example: 100, 224, 340, 664….
- Rule of number 5: A number is divisible by 5 if it ends in 0 or 5. Un número es múltiplo de 5 si acaba en 0 ó 5. Example: 200, 345, 650, 800 …..
- Rule of number 9: A number is divisible by 9 if the sum of its digits is a multiple of 9. Un número es divisible por 9 si la suma de sus cifras es múltiplo de 9. Example: 81, 333, 450, 1278…..
- Rule of number 10: A number is divisible by 10 if it ends in 0. Un número es divisible por 10 si acaba en 0. Example: 30, 400, 500.
- Rule of number 11: A number is divisible by 11 if the difference between the sum of the digits on odd positions and the sum of the digits on even positions is 0, 11 or a multiple of 11. Un número es divisible por 11 si la diferencia entre la suma de las cifras en posición par y la suma de las cifras en posición impar es 0, 11 o un múltiplo de 11. Example: 121, 3652
Solve the following exercises:
- Use the divisibility rules to complete the following table:
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Divisible by |
2 |
3 |
4 |
5 |
9 |
10 |
11 |
25 |
100 |
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375 |
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990 |
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1.848 |
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12.300 |
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14.240 |
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- Find out two numbers with five digits that are divisible by both 2 and 5 and aren’t divisible by 100
- Write down two numbers with five digits that are multiples of:
a) 3 and 11 but not of 9
b) 9 and 11. Are they multiples of 3?
4.- PRIME FACTOR DECOMPOSITION OF A NUMBER
5.- THE HIGHEST COMMON FACTOR AND THE LEAST COMMON MULTIPLE
5.1.- Concept of the highest common factor (HCF)
Definition:
The highest common factor of several numbers is the largest number that evenly divides into all of them.
10.2.- Rule for calculating the h.c.f
Regla:
“To work out the hcf of several numbers, first you have to find the prime factor decomposition of the given numbers and then, to take the common factors with the least index”.
Solve the following exercises:
- Work out the factors of the numbers below and then find out the hcf:
a) 2 and 16 b) 3 and 25 c) 9, 12 and 18 d) 27, 36 and 63
- Find out the hcf of the following numbers using the Spanish and the English methods:
a) 4, 6, 18 and 32 b) 3, 4, 12, 36 and 48
5.3.- Concept of the least common multiple (lcm)
Definition: The least common multiple of several numbers is the smallest number that is multiple of all of them.
5.4.- Rule for calculating the lcm
Regla:
“To work out the lcm of several numbers, first write them as a product of their prime factors and then take the common and non-common factors with the highest index.”
Solve the following exercises:
- Work out the l.c.m. of the numbers below:
a) 9, 12 and 18 b) 27, 36 and 63
- Work out the l.c.m. of the following numbers. What conclusion do you reach?
a) 2, 4, 8 and 16 b) 3, 4, 6 and 12.
- Do this test, don´t copy anything. Call your teacher once you have finished your test:
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