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Example 1: Use mathematical induction to prove that [latex]\large{n^2} + n[/latex] is divisible by [latex]\large{2}[/latex] for all positive integers [latex]\large{n}[/latex]. a) Basis step: show true for [latex]n=1[/latex].
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- Mathematical Induction
$n^2 = 1 \pmod{24}$ for $n=1,5,7,11$, by checking each case individually. $(n+12)^2 = n^2 + 24n + 144 = n^2 \pmod{24}$. Therefore, $n^2 = 1 \pmod{24}$ when $n$ is odd and not divisible by $3$, and so $n^2-1$ is divisible by $24$ for these $n$. You don't need primality of $p$ here!
Divisibility rules are efficient shortcut methods to check whether a given number is completely divisible by another number or not.
How to prove that $p^2-1$ is divisible by $24$ if $p$ is a prime number greater than $3$?
You can use % operator to check divisiblity of a given number. The code to check whether given no. is divisible by 3 or 5 when no. less than 1000 is given below: n=0. while n<1000: if n%3==0 or n%5==0: print n,'is multiple of 3 or 5'. n=n+1.
A divisibility rule is a heuristic for determining whether a positive integer can be evenly divided by another (i.e. there is no remainder left over). For example, determining if a number is even is as simple as checking to see if its last digit is 2, 4, 6, 8 or 0.
Divisibility Rule for 2 and Powers of 2. A number is divisible by if and only if the last digits of the number are divisible by . Thus, in particular, a number is divisible by 2 if and only if its units digit is divisible by 2, i.e. if the number ends in 0, 2, 4, 6 or 8. Proof.
