Problem
NT-B1-M10-P008 Last Two Digits
#8
★★☆☆☆ Level 2 of 5
Find the last two digits of \(9^{37}\).
Look at the cycle of powers of \(9\) modulo \(100\).
We have \(9^1\equiv9\), \(9^2\equiv81\), \(9^3\equiv29\), \(9^4\equiv61\), \(9^5\equiv49\), and \(9^{10}\equiv1\pmod{100}\). Since \(37\equiv7\pmod{10}\), \(9^{37}\equiv9^7\equiv69\pmod{100}\). The last two digits are \(69\).
One may also combine moduli \(4\) and \(25\).