If a = ((13!)^16 (13!)^8)/((13!)^8+(13!)^4) then find the unit digit of…

Key steps: simplify the expression and use the fact that 13! is divisible by 10.

• Let x = (13!)^4. Then a/(13!)^4 = (13!)^16/((13!)^4 + 1) = x^4/(x+1).

• Divide x^4 by x+1: x^4/(x+1) = x^3 - x^2 + x - 1 + 1/(x+1). Thus the integer part (quotient) is x^3 - x^2 + x - 1.

• Note that 13! contains factors 2 and 5, so 13! is divisible by 10 and therefore x = (13!)^4 ends with 0. Hence x, x^2 and x^3 all end with 0.

• Compute the unit digit of the integer part: x^3 - x^2 + x - 1 ≡ 0 - 0 + 0 - 1 ≡ -1 ≡ 9 (mod 10).

Therefore the unit digit of a/(13!)^4 is 9.