(c) According to the question a number is a magic numberv if it satisfies any one of the following conditions: (a) x1x2x3=x4x5x6≠x5x6x7 (b) x1x2x3=x5x6x4≠x4x5x6 (c) x1x2x3=x4x5x6=x5x6x7 Consider ( C )x4x5x6=x5x6x7 implies x4=x5,x5=x6,x6=x7 ⇒x1=x2=x3=x4=............=x7 There are 9 such numbers. The different types of roots and the corresponding number of magic numbers are given below.
x1x2x3
x4x5x6
x7
1
a a a
a a a
a
9
2
a a a
a a a
b
90
10 × 9
3
a a a
b a a
a
90
10 × 9
4
x1x2x3
x1x2x3
—
9900
10 × 990
5
x1x2x3
−x1x2
x3
9900
10 × 9
For (1) a can have any value from 1 to 9. (2,3) a can have any value from 0 to 9 and b can have 9 different values. (4,5)x1x2x3 can have 990(999−9) values and the blank can be filled in 10 ways.