Question

Please answer following question The number of positive integers (a b c d) satisfying 1/a + 1/b + 1/c + 1/d = 1 where a b c d all are distinct

Answer 1

Answer:

Step-by-step explanation:

step 1: if a≥2, then

S=1a+1b+1c+1d<4×12=2. however, S∈N,S>1, which means S≥2, contradiction. i.e.: a=1

step 2: a=1,b>1, assume b≥3 , then

S≤1+13+14+15<1+3×13=2. contradiction. i.e.: b=2

step 3: a=1,b=2. let

Scd=1c+1d,

Scd≤13+14<1

note that S∈N,S>1.

S=11+12+Scd<1+12+1<3

i.e.: S=2

Scd=S−(1+12)=S−32=2−32=12

step 4: c>b=2,c≥3

assume c≥4, then Scd≤14+15<12, contradiction. so, c=3

one may get: d=6

the final result:  a=1,b=2,c=3,d=6