If a and b are positive integers such that a common factor of 2a2 - ab - 3b2 and 2a2 - 5ab + 3b2 is 11, which of the following c

Question

elixir [45]

3 years ago

14

If a and b are positive integers such that a common factor of 2a2 - ab - 3b2 and 2a2 - 5ab + 3b2 is 11, which of the following c

ould equal a?
10
11
12
14
15

Mathematics

1 answer:

Yuki888 [10] · Answer 1 · 2022-03-23T15:57:23+03:00

Factor them
2a^2-ab-3b^2=(2a-3b)(a+b)
2a^2-5ab+3b^2=(2a+3b)(a+b)

notice tat we have the only common factor is
(a+b)
so
a+b=11

we know that it is a POSITIVE integer
that means that it is from 1 to infinty, not including decimals or NEGATIVES
so like
1,2,3,4,5...

let's look at aour options
a+b=11
our choices are
a is
10,11,12
if we try a=10
10+1=11, yah true
if we try a=11
11+0=11
but 0 is not an integer
we can see that if we keep going to 12,14,15, we will get negative integers for b which is not allowed

a=10 is answer