Question

Which of the following are three distinct integers such that the sum of their reciprocals is an integer?

• A. $2,3,4$
• B. $3,4,5$
• C. $5,6,7$
• D. $2,3,6$

Answer 1

The correct option is D $2,3,6$

Option A: $2,3,4$

Sum of their reciprocals $=\frac{1}{2}+\frac{1}{3}+\frac{1}{4}$

L.C.M of $2,3,4=12$

$\therefore$ Sum of reciprocals $=\frac{6+4+3}{12}=\frac{13}{12}$

This is not an integer.

Option B: $3,4,5$

Sum of their reciprocals $=\frac{1}{3}+\frac{1}{4}+\frac{1}{5}$

L.C.M of $3,4,5=60$

$\therefore$ Sum of reciprocals $=\frac{20+15+12}{60}=\frac{47}{60}$

This is not an integer.

Option C: $5,6,7$

Sum of their reciprocals $=\frac{1}{5}+\frac{1}{6}+\frac{1}{7}$

L.C.M of $5,6,7=210$

$\therefore$ Sum of reciprocals $=\frac{42+35+30}{210}=\frac{107}{210}$

This is not an integer.

Option D: $2,3,6$

Sum of their reciprocals $=\frac{1}{2}+\frac{1}{3}+\frac{1}{6}$

L.C.M of $2,3,6=6$

$\therefore$ Sum of reciprocals $=\frac{3+2+1}{6}=\frac{6}{6}=1$

This is an integer.

Hence, option D is correct.