Understanding this proof for the proposition "for all integers a, gcd(9a+4, 2a+1) = 1.

proof: gcd(9a+4, 2a+1) = gcd(2a+1, a) = gcd(a, 1). since gcd(a, 1)=1, gcd(9a+4, 2a+1) =1.

First line because 4(2a+1)=8a+4 and 9a+4-(8a+4)= a

Second line because a times 2 =2a and 2a+1-2a=1

Although the second equality is more or less obvious since 2a+1 leaves a remainder of 1 when divided by a.

180

step-by-step explanation:

reduce the friction between its moving parts

step-by-step explanation:

in fact, by reducing the friction between the moving parts of the machine, it is possible to reduce the energy wasted due to this friction; therefore, more input energy is converted into useful work, and this will improve the efficiency of the machine.