Prime and Composite Numbers. Divisibility – We Solve Problems

#### Prime and Composite Numbers. Divisibility

Among all natural numbers we can distinguish $\textit{prime}$ and $\textit{composite}$ numbers.
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A number is $\textit{composite}$ if it is a product of two smaller natural numbers. For example, $6 = 2\times3$. Otherwise, and if the number is not equal to 1, it is called $\textit{prime}$. The number 1 is neither prime nor composite.
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The well-known fact, called the $\textbf{Fundamental Theorem of Arithmetic}$, says that any natural number greater than 1 can be uniquely expressed as a product of prime numbers in non-decreasing order. For example, $$630=2\times3\times3\times5\times7=2\times3^2\times5\times7$$
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$\textit{Modulo operation}$: Given any two natural numbers $a$ and $b$, called the dividend and the divisor respectively, we can divide $a$ by $b$ with the remainder. That is to find such non-negative integer numbers $c$ and $d$ ($d<b$), called the quotient and the remainder respectively, that $a=c\times b+d$. For example, $41=2\times15+11$ is the division of 41 by 15 with the remainder 11 and $5=0\times7+5$ is the division of 5 by 7 with the remainder 5.
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If $a$ is divided by $b$ with zero remainder (without a remainder) we say that “$a$ is divisible by $b$”$\;$or “$b$ divides $a$”. From the definition of \textit{modulo operation} for $a$ the property to be divisible by $b$ is equivalent to the existence of non-negative integer $c$ such that $a=c\times b$. We denote it by $a÷b$ for “$a$ is divisible by $b$”$\;$and $b|a$ for “$b$ divides $a$”. For example, $105÷7$ and $9|111111111$ because $105=15\times7$ and $111111111=12345679\times9$.
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We immediately deduce from the Fundamental Theorem of Arithmetic that if a product of two natural numbers is divisible by a prime number, then one of these numbers is divisible by this prime number.
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Problems:

#### Divisibility of a number. General properties , Number theory. Divisibility

Is it true that if a natural number is divisible by 4 and by 6, then it must be divisible by $4\times6=24$?

#### Divisibility of a number. General properties , Number theory. Divisibility

And what if a natural number is divisible by 5 and by 7? Should it be divisible by 35?

#### Divisibility of a number. General properties , Number theory. Divisibility

The number $A$ is not divisible by 3. Is it possible that the number $2A$ is divisible by 3?

#### Divisibility of a number. General properties , Divisibility rules , Number theory. Divisibility , Prime numbers

List the first 25 prime numbers. Done? Now write the prime decomposition of 2910.

#### Divisibility of a number. General properties , Divisibility rules , Number theory. Divisibility

Lisa knows that $A$ is an even number. But she is not sure if $3A$ is divisible by 6. What do you think?

#### Divisibility of a number. General properties , Divisibility rules , Number theory. Divisibility

George divided number $a$ by number $b$ with the remainder $d$ and the quotient $c$. How will the remainder and the quotient change if the dividend and the divisor are increased by a factor of 3?

#### Products and factorials , The fundamental theorm of arithmetic. Prime factorisation.

Let us introduce the notation – we denote the product of all natural numbers from 1 to $n$ by $n!$. For example, $5!=1\times2\times3\times4\times5=120$.

a) Prove that the product of any three consecutive natural numbers is divisible by 3!=6. b) What about the product of any four consecutive natural numbers? Is it always divisible by 4!=24?

#### Divisibility of a number. General properties , Number theory. Divisibility

Can a sum of three different natural numbers be divisible by each of those numbers?

#### Divisibility of a number. General properties , Number theory. Divisibility

A young mathematician felt very sad and lonely during New Year’s Eve. The main reason for his sadness ( have you guessed already?) was the lack of mathematical problems. So he decided to create a new one on his own. He wrote the following words on a small piece of paper: “Find the smallest natural number n such that n! is divisible by 2018″$\;$, but unfortunately he immediately forgot the answer. What is the correct answer to this question?

#### Divisibility of a number. General properties , Number theory. Divisibility

Find such a natural number $n$ that all the numbers $n+1$, $n+2$,…, $n+2018$ are composite.

#### Divisibility of a number. General properties , Number theory. Divisibility

The numbers $2^{2018}$ and $5^{2018}$ are expanded and their digits are written out consecutively on one page. How many digits are on the page?

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