Question

The last two digits of the number $(23{)}^{14}$ are

• A. $01$
• B. $05$
• C. $03$
• D. $09$

Answer 1

The correct option is D $09$

$(23{)}^{14}=(529{)}^7=(530-1{)}^7={}^7{C}_0(530{)}^7-{}^7{C}_1(530{)}^6+\dots +{}^7{C}_6(530{)}^1-1={}^7{C}_0(530{)}^7-{}^7{C}_1(530{)}^6+...+3710-1=100k+3709,k\in Z$

Therefore, last two digits are $09$.