A whole number is input into the machine The output is then used as a new input to produce a second output. Rosa says "the last digit of the second output must be a 5." Is Rosa correct? Explain your answer

Let #e# be an even number and put it into the first machine.

#e + 3# will be an odd number.

#e + 3 = e + 2 + 1#

An even number plus #2# will be another even number.

Then, an even number plus #1# will be an odd number.

Therefore #e + 3 = # and odd number#

Let #d# be an even number and put it into the first machine.

#d + 3# will be an even number.

#d + 3 = d + 2 + 1#

An odd number plus #2# will be another odd number.

Then, an odd number plus #1# will be an even number.

Therefore #d + 3 = # an even number#

Therefore Machine 1 can output an even or an odd number

Let #d# be an odd number and put it into the second machine.

#d xx 5 =# a number ending in #5#

An odd number times #5# always ends in #5#

#5 xx 1 = 5#

#5 xx 3 = 15#

#5 xx 5 = 25#

#5 xx 7 = 35#

#5 xx 9 = 45#

Let #e# be an odd number and put it into the second machine.

#e xx 5 =# a number ending in #0#

Because #e# is even it can be written as #2 xx i# where #i# is an integer:

Then:

#5 xx e = 5 xx (2 xx i) = 10 xx i = i0#

Therefore Machine 2 can output a number ending in #0# or #5#