Expected CAT Question

Solution 1:

n can be a 2 digit or a 3 digit number.
Case (I) Let n be a 2 digit number. Let n = 10x + y, where x and y are non-negative integers, Pn = xy and Sn = x + y Now, Pn + Sn = n ∴ xy + x + y = 10x + y ∴ xy = 9xy = 9

There are 9 two digit numbers (19, 29, 29, … ,99) for which y = 9
Case (II) Let n be a 3 digit number. Let n = 100x + 10y + z, where x, y and z are non-negative integers,

Pn = xyz and Sn = x + y + z

Now, Pn + Sn = n xyz + x + y + z = 100x + 10y + z ∴ xyz = 99x + 9y ∴ z = 99/y + 9/x
From the above expression, 0 < x, y < 9 But, we cannot find any value of x and y, for which z is a single digit number. ∴ There are no 3 digit numbers which satisfy Pn + Sn = n Hence, option 4.

Solution 2:

110×130÷(70×30)=6.8 but we can only put complete tiles hence 6