# Cryptarithmetic Division Problem 5

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-------   - 1Q T P A  T T N  x   R-------   - 2Q T P S  T T N  x   P-------  - 3B T Q PTaking the third eqn -  T T N  x   P-------  - 3B T Q PN x P = _P is same has P x N = _P thus, the two possible cases according rulementioned in hack 3 on this page will be -Case 1 - N = {3, 7, 9} P = 5Case 2 - P = 6, and N = {2, 4, 8}Case 1.1 - N = 3 and P = 5.  T T 3  x   5-------B T Q 5Now 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 6Q = {1,6}Taking Q = 1  T T 3  x   5-------B T 1 5For getting 1 at 10's place, possible value for T can beT = {2, 4, 6, 8}For T -> 2  2 2 3  x   5-------B T 1 5Now in 10's position 5 x 2 + 1(carry) = 11.1 carry to 100'sNow again at 100's 5 x 2 + 1 carry = 11. So value for T in row 3is 1. T = 1. But, T value was already assumed to be 2. So this is also rejected.Now, lets take T = 4  T T 3  x   5-------B T 6 5Now, 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 3And for T = 8 it will come out to be 4 so rejectedNow, 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 9ie T = {7, 9}Let's try for 7  7 7 3  x   5-------B T 6 5 units -> 3 x 5 = 15 1 carry10's 7 x 5 +1(carry) = 36 3 carry100's 7 x 5 + 3(carry) = 38So T value is 8 but was assumed to be 7 so rejectedNow lets use T = 9  9 9 3  x   5-------B T 6 5 In this case T will come out to be 9 onlyand B =4. We can find all values from here. Solved !`

PT