Some Number Theory Problems




(Try to do as many as you can; I'll be glad to give hints.)

  1. (Example) Prove that the product of two consecutive numbers is always divisible by 2. The numbers are $n$ and $n+1$. By the division algorithm, $n$ is either of the form $2k$ or $2k+1$. In the first case the product is $(2k)(2k+1)$ which equals $2[(k)(2k+1)]$ so is clearly even. In the second case the product is MATH which is also clearly even.

  2. Prove that the product of 3 consecutive numbers is always divisible by 3. In fact, the product of 3 consecutive numbers is divisible by 6: can you prove this? What do you think is true about the product of 4 consecutive numbers? What about $N$ consecutive numbers?

  3. Show that the sum of an integer and its square is always even.

  4. Prove that the sum of the cubes of any three consecutive positive integers is a multiple of $9$.

  5. Explain why every integer is of the form $3n$, $3n+1$, $3n-1$.

  6. An integer is odd if and only if its square is odd. (Remember that there are two things to prove in an "if and only if" statement.)

  7. Prove (by induction) that

    1. MATH

    2. MATH

  8. (Example) Prove that MATH is divisible by 9 when $n\geq 0$. Procede by induction. The case $n=0$ yields the number 54 which is divisible by 9. Suppose now that the claim is true for some number $n\geq 0$; we will show that it's true for $n+1$; i.e., that 9 divides MATH. We subtract:MATH

    Thus, MATH

  9. Prove (use induction) that:

    1. $18^{n}-1$ is always a multiple of a certain number $K>1$ (Find $K$ and prove it.)

    2. $11^{n}-4^{n}$ is always a multiple of $7$

    3. $7^{2n}+16n-1$ is divisible by 64 for $n>0$

    4. MATH is divisible by 24 for $n\geq 0$.

  10. Prove (use induction) that $(1+x)^{n}\geq 1+nx$ where the integer $n\geq 0$, and $x>-1$ is any number.