Maths circle: Divisibility 1 – We Solve Problems

Maths circle: Divisibility 1

Prime numbers are like atoms – they are the building blocks of integers. A prime is any natural number having exactly 2 positive divisors – 1 and itself. $\\$
One of the most important theorems of mathematics, so important, that it is called the fundamental theorem of arithmetic, is about primes. It says that every natural number can be decomposed into a product of primes in exactly one way (up to a rearrangement). Today, we will explore some of the consequences of that theorem. $\\$ Firstly, a prime decomposition of a product of two numbers is a product of the decompositions of these numbers, and so contains all the primes that appear in both decompositions. Secondly, a number is divisible by another number if and only if all the primes in the decomposition of the divisor also appear in the decomposition of the dividend. We can think about dividing natural number as a smaller number “stealing” some of the prime divisors of the larger number. We will see how this simple fact can help us answer some questions about divisibility.

Problems:

1.

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

a) The number $a$ is not divisible by $3$. Could the number $2a$ be divisible by $3$? $\\$ b) We know that the product $c \times d$ is divisible by a prime $p$. Show that either $c$ or $d$ must be divisible by $p$.

2.

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

The number $b^2$ is divisible by $8$. Show that it must be divisible by $16$.

3.

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

Find a number which: $\\$
a) It is divisible by $4$ and by $6$, is has a total of 3 prime factors, which may be repeated. $\\$
b) It is divisible by $6, 9$ and $4$, but not divisible by $27$. It has $4$ prime factors in total, which may be repeated. $\\$
c) It is divisible by $5$ and has exactly $3$ positive divisors.

4.

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

a) The number $a$ has a prime factorization $2^3 \times 3^2 \times 7^2 \times 11$. Is it divisible by $54$? Is it divisible by $154$? $\\$ b) The prime factorization of the number $b$ is $2 \times 5^2 \times 7 \times 13^2 \times 17$. The prime factorization of the number $c$ is $2^2 \times 5 \times 7^2 \times 13$. Is the first number divisible by the second one? Is the product of these two numbers, $b \times c$, divisible by $49000$? $\\$

5.

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

a) The number $a$ is even. Should $3a$ definitely also be even? $\\$ b) The number $5c$ is divisible by $3$. Is it true that $c$ is definitely divisible by $3$? $\\$ c) The product $a \times b$ is divisible by $7$. Is it true that one of these numbers is divisible by $7$? $\\$ d) The product $c \times d$ is divisible by $26$. Is it true that one of these numbers is divisible by $26$?

6.

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

a) The number $a^2$ is divisible by $11$. Is $a^2$ necessarily also divisible by $121$? $\\$ b) The number $b^2$ is divisible by $12$. Is $b^2$ necessarily also divisible by $144$?

7.

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

a) What is the smallest integer $n$ such that $n!$ is divisible by $990$? $\\$
b) Is $100!$ divisible by $2^{100}$

8.

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

Jack believes that he can place $99$ integers in a circle such that for each pair of neighbours the ratio between the larger and smaller number is a prime. Can he be right?

9.

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

a) Prove that a number is divisible by $8$ if and only if the number formed by its laast three digits is divisible by $8$. $\\$ b) Can you find an analogous rule for $16$? What about $32$?

10.

Divisibility of a number. General properties , Euler's theorem , Number theory. Divisibility

Look at this formula found by Euler: $n^2 +n +41$. It has a remarkable property: for every integer number from $1$ to $21$ it always produces prime numbers. For example, for $n=3$ it is $53$, a prime. For $n=20$ it is $461$, also a prime, and for $n=21$ it is $503$, prime as well. Could it be that thiss formula produces a prime number for any natural $n$?

11.

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

Show that if $n! +1$ is divisible by $n+1$, then $n+1$ must be prime. (It is also true that if $n+1$ is prime, then $n! + 1$ is divisible by $n+1$, but you don’t need to show that!)

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