MrWhite
If integer p is a positive factor of 36 and integer t is a positive factor of 27, which of the following products CANNOT be equal to \(p^2 - t^2\) ?
(A) (12)(6)
(B) (19)(17)
(C) (39)(13)
(D) (45)(27)
(E) (63)(9)
The above would be what I would go with.
But if someone is not able to get the logic immediately, the following are another three ways.
Use options.
A) (12)(6)=(p+t)(p-t)….p+t=12 and p-t=6…..Add the two 2p=18 or p=9 and t=3
9 is a factor of 36 and 3 is a factor of 27.
(B) (19)(17) =(p+t)(p-t)….p+t=19 and p-t=17…..Add the two 2p=36 or p=18 and t=1
18 is a factor of 36 and 1 is a factor of 27.
(C) (39)(13) =(p+t)(p-t)….p+t=39 and p-t=13…..Add the two 2p=52 or p=26 and t=13
26 is NOT a factor of 36 and 13 is also NOT a factor of 27.
(D) (45)(27) =(p+t)(p-t)….p+t=45 and p-t=27…..Add the two 2p=72 or p=36 and t=9
36 is a factor of 36 and 9 is a factor of 27.
(E) (63)(9) =(p+t)(p-t)….p+t=63 and p-t=9…..Add the two 2p=72 or p=36 and t=27
36 is a factor of 36 and 27 is a factor of 27.
C is not possible
Numbers being small, write the factors 36: 1*36=2*18=3*12=4*9=6*6
27: 1*27=3*9
36=> 1,2,3,4,6,9,12,18,36
27=> 1,3,9,27
You cannot make both 39 and 13 from p and t.
Another logical wayWhat you have is (p-t) and (p+t).
Add both to get p-t+p+t = 2p. So when you take half of the sum of the two integers, it should be a factor of 36
(A) (12)(6) => \(\frac{12+6}{2}\)=9…yes, 9 is a factor of 36
(B) (19)(17) => \(\frac{19+17}{2}\)=18…yes, 18 is a factor of 36
(C) (39)(13) => \(\frac{39+13}{2}\)=26…NO, 26 is not a factor of 36
(D) (45)(27) => \(\frac{45+27}{2}\)=36…yes, 36 is a factor of 36
(E) (63)(9) => \(\frac{63+9}{2}\)=36…yes, 36 is a factor of 36
C