Question

Fill in the empty squares using numbers from 1
to 9, so that each row and each column has the
right product
All the numbers from 1 to 9 must be used.
Answer Mx x = 54
96
180
2= 200​

Answer 1

Given :  _ _x_=_ _+_ _=_ _       ( 9 blanks paces )

To find :   Blank spaces to be filled  with 1 to 9  Digits

Solution:

Complete Question

 AB

 * C

____

 DE

+ FG

______

 HI

Here  AB * C  = DE  ( is one operation)

DE + FG  =  HI    ( another operation)

there is no linkage directly between AB & HI

A to I digit 1 to 9

17 x  4  = 68  +  25  = 93

17 * 4  = 68

68 + 25 = 93

 17

 * 4

____

 68

+ 25

______

 93

Here is how we reached to solution

All digits used from 1 to 9

ab* c  = de

de + fg = hi

1 to 9 digit

hi two digit

b , c can not be 1  or 5

as ab * 1 = ab => ab = de ( with same digits)

a1 * c will end with c   a5 * c  will end with 5 or 0

ab * 5 will end with 0 or 5   ( 0 not available , 5 will be repeated digit in c & i)

Let s check all possible

11 , 12 * 1 , 12 * 2 ,  12 * 3 = 36 , 12 *  4 = 48 , 12 * 5 ,12*6 = 72 , 12 * 9 = 108 ( not possible)

12 * 7 = 84  remaining digits 3 , 5 , 6  9   35 + 69 = 104  ( not satisfied)

12 * 8 = 96 ( remaining digits 3 , 4 , 5 , 7 ) not  satisfied

13 * 1 , 13 * 2 = 26 , 13 * 3 , 13 * 5 , 13 * 7 = 91 , 13 * 8 = 104  ( not possible)

13 * 4 = 52 remaining digits 6 , 7 , 8 . 9 not  satisfied

13 * 6 = 78  remaining digits 2 , 4 , 5 . 9  not  satisfied

14 * 1, 14 * 2 = 28 , 14 * 3 = 42 , 14 * 4 , 14 * 5 , 14 * 6 = 84 , ( not possible)

14 * 7  = 98  remaining digits 2 , 3 , 5 . 6  not  satisfied  

15 , 16 * 1 , 16 * 2 = 32 , 16 * 4 = 64 , 16 * 5 . 16*6 , 16 * 7 = 112 ( not possible)

16 * 3  = 48   remaining digits 2 , 5 , 7 , 9   not  satisfied

17 * 1 , 17*3 = 51 , 17 * 5 , 17 * 6 = 102 ( not possible)

17 * 2 = 34   remaining digits 5 , 6 ,  8 , 9 not  satisfied

17 * 4 = 68   remaining digits 2 , 3 ,  5 , 9  (68 + 25 = 93)

we got 1st Solution .

18 * 1 , 18 * 5 , 18 * 6 = 108 ( not possible)

18 * 2 = 36  remaining digits  4 5 , 7 9  not  satisfied

18 * 3 = 54 remaining digits  2 6 , 7 9  not  satisfied

18 * 4 = 72 remaining digits  3 5 , 6 9  not  satisfied   36 + 59 = 95

19 * 1 , 19 * 5  , 19 * 6 = 114   ( not possible)

19 * 2 = 38   remaining digits  4 , 5 , 6  , 7   not  satisfied

19 * 3 = 57   remaining digits  2 , 4 , 6  , 8   not  satisfied

19 * 4 = 76 remaining digits  2 , 3 , 5  , 8   not  satisfied

21 , 22 , 23 * 1 , 23 * 2 , 23 * 3 , 23 * 4 = 94 , 23 * 5 , 23 * 6 = 138 ( not possible)

24* 1 , 24 * 2 , 24 * 3 = 72 , 24 * 4  , 24 * 5  ,  25( not possible)

26 * 1 , 26 * 2 , 26 * 4 = 104 ( not possible)

26 * 3 = 78   remaining digits   1 , 4 , 5 , 9   not  satisfied

27 * 1 , 27 * 2 , 27 * 4 = 108 ( not possible)

27 * 3 = 81      remaining digits  4 ,5 , 6 , 7   not  satisfied

28 * 1 , 28 * 2 , 28 * 3 = 84  ( not possible)

29 * 1 , 29 * 2  , 29 * 4 = 116 ( not possible)

29 * 3 = 87   remaining digits  1 , 4 , 5 , 6    

31 , 32 *1  , 32 * 2 , 32 * 3 , 32 * 4 = 128  ( not possible)

33 , 34 * 1 , 34 * 3 , 34 * 4  (136) ( not possible)

35 , 36 * 1 , 36 * 2 = 72  36 * 3 = 108 ( not possible)

37 * 1 , 37 * 2 = 74 , 37 * 3 = 111 ( not possible)

38 * 1 , 38 * 3 = 114 ( not possible)

38 * 2 = 76       remaining digits   1 , 4 , 5 , 9

39 * 1 , 39 * 3 = 117  ( not possible)

39 * 2 = 78   remaining digits   1 , 4 , 5  , 6   ( not satisfied)

41 , 42 * 1  , 42 * 2 , 42 * 3 = 126 ( not possible)

43 * 1 , 43 * 3 = 129 ( not possible)

43 * 2 = 86   remaining digits 1 , 5 , 7  , 9    

44 , 45 , 46 * 1 ,  46 *2 = 92 , 46 * 3 = 138  (not possible)

47 * 1 , 47 *2 = 94 , 47 * 3 = 141  ( not possible)

48 * 1  , 48 * 3 = 144 not possible

48 * 2 = 96  remaining digits  1 , 3 , 5 , 7    

49 * 1   , 49 * 2 = 98 , 49 * 3 = 147 ( not possible)

not further possible as will lead to 3 digit number

17 x  4  = 68  +  25  = 93

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