Exercises - Divisibility

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  1. What can you conclude if $a$ and $b$ are nonzero integers such that $a \mid b$ and $b \mid a$?

  2. Show that if $a$, $b$, $c$, and $d$ are integers with $a$ and $c$ nonzero, such that $a \mid b$ and $c \mid d$, then $ac \mid bd$.

  3. Are there integers $a$, $b$, and $c$ such that $a \mid bc$, but $a \nmid b$ and $a \nmid c$? What can you say about the nature of such integers $a$ if they exist?

  4. Show that if $a$, $b$, and $c \neq 0$ are integers, then $a \mid b$ if and only if $ac \mid bc$

  5. Show that if $a$ and $b$ are integers such that $a \mid b$, then $a^n \mid b^n$ for every positive integer $n$

  6. Show that if $a \mid b$ and $a \mid c$, then $a \mid (bx + cy) \quad \forall x,y \in \mathbb{Z}$

  7. An "even" number is a number of the form $2n$, where $n$ is an integer. An "odd" number is one of the form $2n+1$, where $n$ is an integer. Show that the sum of two even or two odd numbers is even, the sum of an odd and an even integer is odd, and the product of two odd numbers is odd.

  8. Prove that the product of two integers of the form $4k+1$ is again of this form, while the product of two integers of the form $4k+3$ is of the form $4k+1$.

  9. Prove that no integer in the following sequence is a perfect square: $$11, 111, 1111, 11111, \ldots$$

  10. Show that the square of every odd integer is of the form $8n+1$ and the fourth power of every odd integer is of the form $16k+1$

  11. Show that the square of any integer is either of the form $3n$ or $3n+1$.

  12. Show that the product of two integers of the form $6k+5$ is of the form $6k+1$.

  13. Show that if $a$ is an integer, then 3 divides $a^3-a$.

  14. Prove that the product of any three consecutive integers is divisible by 6.

  15. Show that if $a$ and $b$ are both odd, then 8 divides $a^2-b^2$.  

  16. Show that if $a$ is odd, then 12 divides $a^2 + (a+2)^2 + (a+4)^2 + 1$.  

  17. Prove each of the following for a given positive integer $n$:  

    1. $5 \mid n$ if and only if the last digit of $n$ is a $0$ or $5$
    2. $9 \mid n$ if and only if the sum of the digits of $n$ is divisible by $9$
    3. $3 \mid n$ if and only if the sum of the digits of $n$ is divisible by $3$
  18. To find the alternating sum of the digits of a positive integer, we start with zero, and then add the first digit, subtract the second, add the third, subtract the fourth, and so on, alternating between adding and subtracting until we terminate with the last digit of the number. For example, alternating sum of the digits of $12345$ is $1 - 2 + 3 - 4 + 5$. Likewise, the alternating sum of the digits of $9876$ is $9 - 8 + 7 - 6$. Show that a positive integer is divisible by $11$ if and only if its alternating sum is divisible by $11$.

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