Given:

N = 0.738738738...

M = 0.531531531...

Concept:

Non-terminating repeating decimal or recurring decimal:-

A decimal fraction in which a figure or group of figures is repeated indefinitely, as in 0.7777 or as in 1.445445445.

Ex- 0.333... = 0.3̅  = 3/9

Calculation:

N = 0.738738738... = \(0.\overline {738} = \frac{{738}}{{999}}\)

M = 0.531531531... = \(0.\overline {531} = \frac{{531}}{{999}}\)

⇒ N = 738/999 = 82/111

⇒ M = 531/999 = 59/111

Now,

(1/N) + (1/M) = (111/82) + (111/59)

\( ⇒ \frac{{111\: ×\: 59\:+ \:111\: × \:82}}{{82\; ×\: 59}}\)

\(⇒ \frac{{111\left( {59\; + \;82} \right)}}{{82\; ×\: 59}} = \;\frac{{111\; × \;141}}{{82\; × \:59}}\)

⇒ 15651/4838

Shortcut:

Given:

N = 0.738738738...

M = 0.531531531...

Calculation:

N = 0.738738738... = \(0.\overline {738} = \frac{{738}}{{999}}\)

M = 0.531531531... = \(0.\overline {531} = \frac{{531}}{{999}}\)

⇒ N = 738/999 = 82/111

⇒ M = 531/999 = 59/111

(1/N) + (1/M) = (111/82) + (111/59)

unit digit of dominator = 2 and 9 = 2 × 9 = 18

∴ we check option which unit digit of dominator is 8, that is my answer. In

option only option b which unit digit is 8 so that is my answer.