Contents

A. Divisibility

• Múltiplos de un número. Divisores de un número. / Multiples of a number. Divisors of a number. Criterios de divisibilidad: 2, 3, 4, 5, 6, 9, 10, 25, 100 y 11. / Divisibility rules.
• Números primos y números compuestos / Prime numbers and composite numbers.
• Descomposición de un número en factores primos / Prime factor decomposition of a number.
• Máximo común divisor de dos o más números. Mínimo común múltiplo de dos o más números. / Greatest common divisor (GCD) or Greatest common factor (GCF). Least common multiple (LCM).

B. Integers

• Números enteros. Representación. Valor absoluto y opuesto / Integers. Integers as points in a number line. Absolut value. Opposite.
• Suma y resta de números enteros / Addition and substraction of integers.
• Multiplicación y división de números enteros / Multiplication. Rules of the signs. Division.

C. Powers and Square roots (click the link).

3. Find the biggest number smaller than 100 that is:

1. a multiple of 2
2. a multiple of 3
3. a multiple of 4
4. a multiple of 5
5. a multiple of 6
6. a multiple of 7

4. Find the smallest number that is bigger than 1000 that is:

1. a multiple of 6
2. a multiple of 7
3. a multiple of 8

5. Remember with this example: 3 is a is a factor or divisor of 15 because 15 : 3 = 5 (exact division). If 15 is a multiple of 3, then 3 is a divisor of 15. Given that, find all the divisors of 15, 16, 27, 44, 13 and 37.

6. Example: D(15) = {1, 3, 5, 15} Look, 1 and 15 are also divisors. When you have found one factor (3), there is always another factor that goes with it (5) –unless the factor is multiplied by itself to give the number.

7. What does happen to 13 and 37?

Have a look at this divisibility rules page and then, do the test at the bottom.

9. Which of the following numbers is a prime one: 21, 82, 31, 33.

10. Safari fotográfico: Múltiplos de 37. Lee el artículo enlazado. Busca múltiplos de 37 en la vida real y hazles una foto con tu móvil. Ojo, no vale "cortar" números para que sean múltiplos de 37, ni prepararlos (por ejemplo, escribiéndolos en una pizarra). Tienen que ser múltiplos "salvajes". Se valorarán más los números más altos y las mejores fotos a criterio del profesor.

12. Find these numbers:

• Written in order, the four factors of this number make a number pattern in which each number is twice the one before
• An odd number that is a multiple of 7.

13. Find the prime factors of 36, 100, 24, 98, 180, 120, 510 and 640.

14. Aplica los criterios de divisibilidad para comprobar si el número 515902 es divisible entre 2, 3, 4, 5 y 11.

15. Halla todos los divisores de 108.

Mínimo común múltiplo y máximo común divisor

1. Explain what happen in this comic strip and if there is a better solution.
2. If hot-dog sausages are sold in packs of 10 and hot-dog buns are sold in packs of 8, how many of each must you buy to have complete hot dogs with no extra sausages or buns?
3. A bell chimes every 6 seconds. Another bell chimes every 5 seconds. If they both chime together, how many seconds will it be before they both chime together again?
4. Fred runs round a running track in 4 minutes. Debbie runs round in 3 minutes. If they both start together on the line at the end of the finishing straight, when will they both be on the same line together again? How many laps will Debbie have run? How many laps will Fred have run?
5. Find the LCM (Least Common Multiple) of these pairs of numbers. (Write down the smallest number, in prime factor form, that includes all the prime factors of both). 24 and 56, 21 and 35, 12 and 28, 28 and 42, 12 and 32, 18 and 27, 15 and 25, 16 and 36, 96 and 84.
6. Find the GCD (Great Common Divisor) of these pairs of numbers (Write down, in prime factor form, the biggest number that is in the prime factors of both): 12 and 28, 12 and 32, 12 and 35, 13 and 90, 15 and 25, 16 and 36, 18 and 27, 21 and 35, 24 and 56, 24 and 102, 25 and 35, 28 and 42, 42 and 21, 48 and 64, 36 and 54, 96 and 84.
7. Antonio está haciendo una colección de cromos. Los cromos se venden en sobres con 5 cromos. ¿Puede comprar 15 cromos? ¿y 17? ¿Cuántos sobres tiene que comprar como mínimo si la colección consta de 180?
8. David tiene una bolsa con canicas. Hay entre 40 y 80. Si las agrupa de 3 en 3, de 4 en 4 o de 5 en 5, nunca le sobra ninguna. ¿Cuántas canicas hay en la bolsa?
9. Andrew is 15 years old, Carmen is 28, Benito, 45 and George, 39. Knowing that Andrew's father age is a multiple of his son's one, Who's Andrew's father?
10. Bill sold his scooter to Tom for $100. Tom sold it back to Bill for $80. Then Bill sold it to Herman for $90. What is Bill's total profit? (Source: Martin Gardner).
11. How many odd - three digits - multiples of three can you make with the digits of 1234?
12. Intenta calcular el mínimo común múltiplo de 24 y 35 "a lo bruto", esto es, escribiendo la ristra de múltiplos de uno y otro número hasta que coincida el primero. ¿Qué ocurre?
13. En la final del World Padel Tour se enfrenta la pareja Fernando Belasteguín y Pablo Lima contra Sanyo Gutiérrez y Paquito Navarro. En el tie-break van 256 a 257 pero tras una discusión, no saben a quien le toca el servicio. Cada jugador sirve dos veces y empezó Belasteguín que, como establecen las normas del tie-break, comenzó sacando un solo servicio. ¿A quién le tocar sacar ahora?
14. Un padre y un hijo son marineros. Salen juntos el primer día que el hijo se hace a la mar en diferentes barcos de pesca. El padre vuelve a casa cda 20 días y el hijo lo hace cada 15 días. ¿Cada cuánto tiempo coinciden en su casa? (Problema de Olimpiada).
15. El número romano Zipi = CCXX tiene un gran amigo que, obviamente, se llama Zape. Zape es el gran amigo de Zipi porque es la suma de todos los divisores de Zipi (sin contar al propio Zipi). ¿Podrías averiguar qué número es Zape y cómo se expresa en números romanos? Comprueba también que Zipi es el gran amigo de Zape porque es la suma de todos sus divisores (excepto Zape). (Problema de Olimpiada).

Para ampliar: 5 grandes enigmas de las mates.

Integers

Which is the biggest negative number than appear in the video?

What don't like math teachers?

Write a comparison between numbers that appear in the video.

Look at the number line: negatives to the left of 0 and positives to the right of it.

Knowing that:

Numbers to the right of any number on the number line are always bigger (or larger, greater) than that number and

Numbers to the left of any number on the number line are always smaller than that number. You can use also the symbols > (more than) or < (less than) to compare numbers.

1. Fill the gaps:

... is smaller than 4; ... is bigger than –3; 1 > ... ; ... is smaller than –2; –7 is smaller than ..., but bigger than ... ; –1 ... 3; 3 < ... ; ... is smaller than 0 and bigger than –2 ; –4 > ... ; ... is 5 units greater than 1 ; ... is 3 units smaller than –1 ; –8 > ... ; 0 ... –4 ; –2 ... –4

2. The temperatures on three winter days are 1 °C, –4 °C and –2 °C. Write down these temperatures, in order, with the lowest first. What is the difference in temperature between the coldest and the hottest days?

3. One winter morning, the temperature went up from –3 °C to 2 °C. By how many degrees did the temperature rise?

4. In the afternoon, the temperature fell by six degrees from 2 °C. What was the temperature at the end of the afternoon?

5. Temperatures are recorded at midday in five towns: Penistone (-5º), Huddersfield (+3º), Rotherham (-1º), Kiveton (-3º), Anston (0º). Which town was the coldest? What was the difference in temperature between the coldest and the warmest town?

6. Recuerda que cuando sumas o restas números enteros, no importa el orden en el que lo hagas. Eso se llama propiedad asociativa. Mira este ejemplo: 3 – 5 + 10 – 7 + 2 – 3

• FORMA 1 DE RESOLVERLO ("DEL TIRÓN"): (3 – 5) + 10 – 7 + 2 – 3 = (– 2 + 10) – 7 + 2 – 3 = (8 – 7) + 2 – 3 = (1 + 2) – 3 = 3 – 3 = 0
• FORMA 2, POSITIVOS POR UN LADO Y NEGATIVOS POR OTRO:

3 + 10 + 2 – 5 – 7 – 3 = 15 – 15 = 0

• FORMA 3, A TU AIRE: 3 – 5 + 10 – 7 + 2 – 3 = 0 + 0 = 0

7. Work out the following:

A) 7 + 3 – 5 = B) –2 + 3 – 7 = C) –1 + 3 + 4 = D) –2 – 3 + 4 = E) –1 + 1 – 2 = F) –4 + 5 – 8 = G) –3 + 4 – 7 = H) 1 + 3 – 6 = I) 8 – 7 + 2 – 5 – 7 + 12 = J) –4 + 5 – 8 – 4 + 6 – 8 = K) 203 – 202 + 7 –1 + 4 – 2 = L) –6 + 9 – 12 –3 – 3 – 3 = M) –3 + 4 – 6 –102 + 45 – 23 = N) 8 – 10 – 5 + 9 – 12 + 2 + 99 – 100 – 46 = Ñ) 2547 + 3899 – 1885 – 2546 – 3898 = O) 12 - 33 - 15 - 21 + 43 =

8. A sequence begins: 4, 1, –2, –5 … Find out the following number of the sequence. Write down the rule for this sequence.