You may have seen almost similar problem in the examples previously, this is a harder version of the same problem and the way of solving is different. We will suggest not to look at the solution even once, keep this page opened and try solving yourself and if you get stuck somewhere then see the hints.

We shall first try to convert this into multiple Multiplication problems –

PT

Converting into Multiplication problem T T N x R ------- - 1 Q T P A

T T N x R ------- - 2 Q T P S

T T N x P ------- - 3 B T Q P

Taking the third eqn -

T T N x P ------- - 3 B T Q P

N x P = _P is same has P x N = _P thus, the two possible cases according rule mentioned in hack 3 on this page will be -

Case 1 - N = {3, 7, 9} P = 5 Case 2 - P = 6, and N = {2, 4, 8}

Case 1.1 - N = 3 and P = 5.

T T 3 x 5 ------- B T Q 5

Now at unit's 3 x 5 = 15 thus 1 is carry to next operation. Now T x 5 + 1(carry) possible values are -> {1, 6}. as multiplication with 5 will give 0 or 5 and add 1 to both so 1 and 6

Q = {1,6} Taking Q = 1

T T 3 x 5 ------- B T 15

For getting 1 at 10's place, possible value for T can be T = {2, 4, 6, 8}

For T -> 2

223 x 5 ------- B T 15

Now in 10's position 5 x 2 + 1(carry) = 11. 1 carry to 100's Now again at 100's 5 x 2 + 1 carry = 11. So value for T in row 3 is 1. T = 1. But, T value was already assumed to be 2. So this is also rejected.

Now, lets take T = 4

T T3 x 5 ------- B T 65

Now, here again if you solve like previous. T value in row 3, will come out to be 2. But, T value was assumed to be 2, so rejected.

similarly for T = 6 it will be rejected as T will come out as 3 And for T = 8 it will come out to be 4 so rejected

Now, lets try with Q = 6.(Q had two values 1 and 6)

To get 6 at tens place T value can only be 7 or 9 ie T = {7, 9}

Let's try for 7 773 x 5 ------- B T 65

units -> 3 x 5 = 15 1 carry 10's 7 x 5 +1(carry) = 36 3 carry 100's 7 x 5 + 3(carry) = 38

So T value is 8 but was assumed to be 7 so rejected

Now lets use T = 9

993 x 5 ------- B T 65

In this case T will come out to be 9 only and B =4. We can find all values from here. Solved !

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