Question

M=100a+10b+c
1:$\frac{100a+10b+c}{1000}$=a+b+c
Help me to answer: a=? ;b=? ;c=? ; M=?
Thanks, 

Answer 1

It appears as if this is a number , integer problem.

M = 100 a + 10 b + c
It  appears like M is a three digit number.  Hence, a ≠ 0. 
       1 ≤ a ≤ 9,   0 ≤ b ≤ 9,   0 ≤ c ≤ 9.
    
  Then:      1 ≤ a+b+c ≤ 27      ---- (1)
                 100 ≤ M ≤ 999       ---- (2)

Given:    1 :  M/1000 =  a+b+c
=>  1000 / M = a+b+c
=>  (a+b+c) * (100 a+ 10 b + c) = 1000 = 1 * 2³ * 5³      --- (3)

Factors of 1000:  1 * 1000, 2*500, 4*250, 8*125, 10*100, 20*50, 40*10....

From equations (1), (2), and (3), one factor of 1000 should be less than 27. Another factor should be more than or equal to100.

Possibilities:     factors:   1 * 1000, 2 * 500, 4 * 250, 8 * 125, 10 * 100
 
Among the above, sum of digits (a,b,c) of the larger factor should be the smaller factor.   So   8 * 125 is correct.

so   a+b+c = 8
       1000a + 10 b + c = 125 = 100 * 1 + 10 * 2 + 5

        =>  a = 1,   b = 2,   c = 5,   M = 125