Question

the units digit if the number (2 power 3 power 4+3 power 4 power 5+4 power 5 power 6 +6 power 7 power 8+9 power 10 power 11)

Answer 1

Try using the equation editor. Its not really clear what the number we are working with is.

Answer 2

Correct?:\[2^{3^4}+3^{4^5}+4^{5^6}+5^{6^7}+6^{7^8}+7^{8^9}+8^{9^{10}}+9^{10^{11}}\]

Answer 3

Do you have any ideas? What tools are you working with? Any theorems?

Answer 4

Like for example, maybe you should start with just one of the terms instead of the whole sum. Have you thought about what the units digit of \(6^{7^8}\) is?

Answer 5

\(5^{6^7}\) and \(6^{7^8}\) will be the easiest ones out of the bunch. Then try the 9 and 4. Once you see the idea, the rest should follow.

Answer 6

Yeah, you only need to solve for the units digit of each term. Each power will follow predictable patterns modulo 10.

Answer 7

For instance, the powers of 2 are 2,4,8,16,32,64,128,256,512,1024,... but if we just look at the units digit, it repeats: 2,4,8,6,2,4,8,6,2,4,8,6,...