What an odd fact(or)
Problem
Getting Started
What can you say about the patterns in the last digits of powers of $2$, $3$, $4$ etc?
How can you use these patterns to say what the last digits are of the numbers raised to the power $99$?
Now can you say whether the sum is divisible by $5$?
Student Solutions
Thanks to Steve of Bayridge Secondary School for the following solution. Solutions were also recieved from Hin-Tai from Bourne Grammar School, Junwei from BHASVIC, Josh of Lawrence High School, Stephanie and Stephen of Knowles Hill School, Andrei of School 205, Bucharest and Robert and Josh from Highgate school.
Well done to all of you but there is more. What powers and what numbers will this pattern work for? Charlotte of The Mount School, York thinks it works for all powers.
Show that $1^{99} + 2^{99} + 3^{99} + 4^{99} + 5^{99}$ is divisible by 5.
For each base (1, 2, 3, 4 or 5), there exists a pattern between
the value of the exponent, and the first place value.
For $n ^ 1$, the first place value is always 1.
|
n
|
$1 ^ n$ | $2 ^ n$ | $3 ^ n$ | $4 ^ n$ | $5 ^ n$ |
|
1
|
1
|
2
|
3
|
4
|
5
|
|
2
|
1
|
4
|
9
|
16
|
25
|
|
3
|
1
|
8
|
27
|
64
|
125
|
|
4
|
1
|
16
|
81
|
256
|
625
|
|
5
|
1
|
32
|
243
|
...4
|
...5
|
|
6
|
1
|
64
|
729
|
...6
|
...5
|
|
7
|
1
|
128
|
...7
|
...4
|
...5
|
|
8
|
1
|
256
|
...1
|
...6
|
...5
|
if (n mod 4) = 0 then $2 ^ n$ mod 10 = 6
if (n mod 4) = 2 then $3 ^ n$ mod 10 = 9
if (n mod 4) = 3 then $3 ^ n$ mod 10 = 7
$4 ^ n$
if (n mod 2) = 0 then $4 ^ n$ mod 10 = 6
if (n mod 2) = 1 then $4 ^ n$ mod 10 = 4
$5 ^ n$
$5 ^ n$ mod 10 = 5
Let n = 99
n mod 4 = 3
n mod 2 = 1
$1^{99}$= 1
$2^{99}$ = ...8
$3^{99}$ = ...7
$4^{99}$ = ...4
$5^{99}$ = ...5
-------------
...15
Let $m$ be a natural number greater than, or equal to 5.
$\{m \ \mid \ m \geq 5, m \in N \}$
if m mod 10 = 0, 5, then m is divisible by 5Example:
5 mod 5 = 0
90 mod 5 = 0
5 mod 10 = 5
90 mod 10 = 0
$1^{99} + 2^{99} + 3^{99} + 4^{99} + 5^{99} mod 10 = 5$
therefore $1^{99} + 2^{99} + 3^{99} + 4^{99} + 5^{99}$ is divisible by 5.