If-b-Z-both-the-roots-of-equation-x-2-bx-132-0-are-integers-Find-1-the-largest-possible-value-of-b-2-the-smallest-possible-value-of-b-

Question Number 111035 by ZiYangLee last updated on 01/Sep/20

If b \in Z^{+} \forall both the roots of equation

x^{2} - b x + 132 = 0 are integers .

Find

(1) the largest possible value of b

(2) the smallest possible value of b .

Answered by 1549442205PVT last updated on 02/Sep/20

x^{2} - bx + 132 = (x - m) (x - n)

\Leftrightarrow {\begin{cases} m + n = b > 0 \\ mn = 132 = 3.11.4 \end{cases}

WLOG suppose m ⩽ n

\Rightarrow m \in {1, 2, 3, 4, 6, 11}

i) m = 1 \Rightarrow n = 132 \Rightarrow m + n = 133

ii) m = 2 \Rightarrow n = 66 \Rightarrow m + n = 68

iii) m = 3 \Rightarrow n = 44 \Rightarrow m + n = 47

iv) m = 4 \Rightarrow n = 33 \Rightarrow m + n = 37

v) m = 6 \Rightarrow n = 22 \Rightarrow m + n = 28

vi) m = 11 \Rightarrow n = 12 \Rightarrow m + n = 23

Thus, b = m + n get smallest value

equal to 23 so that the given eq .

x^{2} - bx + 132 has both its roots are

integers

Commented by 1549442205PVT last updated on 02/Sep/20

Since m + n = b > 0, but mn = 132, so

both are positive, Sir!

Commented by Her_Majesty last updated on 01/Sep/20

m > 0 \land n > 0 o r m < 0 \land n < 0 \Rightarrow c h e c k y o u r

a n s w e r . t h e m e t h o d i s o k

Commented by Her_Majesty last updated on 02/Sep/20

s o r r y I m i s s e d t h a t b \in Z^{+}, y o u a r e r i g h t