Long division?

Number Theory Level 3

Given that m − 2 m - 2 is a positive integer that divides 3 m 2 − 2 m + 11 3m^2 - 2m + 11 , find the sum of all such values of m m .


The answer is 24.

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2 solutions

3 m 2 − 2 m + 11 = ( 3 m + 4 ) ( m − 2 ) + 19 3m^{2} -2m +11 = (3m+4)(m-2) + 19
Since ( m − 2 ) ∣ 3 m 2 − 2 m + 11 (m-2) | 3m^{2} - 2m+11
( m − 2 ) ∣ 19 (m-2) | 19
( m − 2 ) k = 19 (m-2)k = 19
m − 2 = 1 , m − 2 = 19 m-2= 1 , m-2= 19
m = 3 , 21 m = 3, 21


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Topalli Murti
Feb 21, 2019

(m-2)(3m+4+19/(m-2))=3m^2-2m+11

Since m-2 divides 3m^2-2m+11, 19/(m-2) has to be an integer. The only values of m that fit this are 21 and 3 so the answer is 24.

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