### More Mods

What is the units digit for the number 123^(456) ?

### Novemberish

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

### A Biggy

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

# Negative Power

##### Stage: 4 Challenge Level:
• What does $4$ to the power $3$ mean ?
• What does $4$ to the power $-3$ mean ?
• What does $-4$ to the power $3$ mean ?
• What does $-4$ to the power $-3$ mean ?
Can you see two interpretations that might be made of the expression : $4$ to the power $3$ to the power $2$ ?

One of those interpretations comes to the same as $4$ to the power $6$, and the other interpretation would make $4$ to the power $9$. Go with the interpretation that makes $4$ to the power $6$.