Question

1).Sum of the digits of a two-digit number is 10. When we interchange the digits, it is
found that the resulting new number is greater than the original number by 28. What are the numbers .
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Answer 1

Step-by-step explanation:

There is a simple test for divisibility by 9 that states a number is divisible by 9 if (and only if) the (repeated) digit sum of the number is divisible by 9. For example, 18 and 81 are divisible by 9 because 1+8=9.

So the two-digit number we are looking for is divisible by 9.

Moreover the difference to another number divisible by 9 is 27. For the majority of numbers (those ending in 4 and higher), adding 27 lowers the units place by 3 so I started thinking about those, and quickly found 36. But a more structured way is nothing that the two-digit multiples of nine pair up:

18 <=> 81

27 <=> 72

36 <=> 63

…

And of course, the difference is a multiple of 9 too, decreasing in steps of 2 because you increase one and decrease the other number by 9 each time:

81 - 18 = 9 x 9 - 2 x 9 = 7 x 9

72 - 27 = (9–1) x 9 - (2+1) x 9 = 5 x 9

63 - 36 = 3 x 9

54 - 45 = 1 x 9

…

Now of course, it's no coincidence that 27 = 3 x 9 so starting from either side of the above list quickly finds (36, 63) without solving any equations!

And note that even though the question doesn't immediately generalize to three and more digit numbers, the ideas about divisibility and differences continue to hold even when you cannot write out all possibilities so easily.