Tenth power
Do these powers look odd...?
Age
11 to 14
Challenge level
Problem
Which of the following three statements are true?
a) $ 3^{10} $ is even
b) $ 3^{10} $ is odd
c) $ 3^{10} $ is a square number
b) $ 3^{10} $ is odd
c) $ 3^{10} $ is a square number
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: (b) and (c) are true, (a) is not ($3^{10}$ is odd and square)
$3^{10}=3\times3\times3 ...\times3$ product of odd numbers $\therefore$ odd (and not even)
$\therefore$ (a) is not true but (b) is
$3^{10} = \left( 3^5 \right)^2$, so it is also a perfect square $\therefore$ (c) is true
(because $3^{10}=\underbrace{3\times3\times...\times3}_{\text{10 times}} = \underbrace{3\times3\times...\times3}_{\text{5 times}}\times\underbrace{3\times3\times...\times3}_{\text{5 times}} = \left(3^5\right)\times\left(3^5\right)$)