Albert said:$(1)A,B \in N,A-B=98$
(2)All the digit-sum of $A$ and $B$ are multiple of 19
please find the smallest of $A\,\, and \,\, B$
The equation for finding the smallest value of A and B is A = 98 + 19n and B = 19n, where n is any positive integer. This equation satisfies the condition that A-B=98 and their digit-sum is a multiple of 19.
We can prove this by considering the properties of the digit-sum. The digit-sum of a number will always be less than or equal to the number itself. Therefore, by setting B = 19n, we are ensuring that the digit-sum of B is the smallest possible value. Since A is defined as A = 98 + 19n, the digit-sum of A will be larger than the digit-sum of B, but still a multiple of 19. Hence, this equation gives the smallest possible values for A and B.
Yes, this equation can work for any difference that is a multiple of 19. For example, if the difference is 38, the equation will be A = 38 + 19n and B = 19n. The digit-sum of A and B will still be a multiple of 19.
If we use a negative difference, the equation will still hold true. However, the values of A and B will be negative as well. For example, if the difference is -98, the equation will be A = -98 + 19n and B = 19n. The digit-sum of A and B will still be a multiple of 19.
No, this equation only works when the difference is a multiple of 19 and the digit-sum is also a multiple of 19. If these conditions are not met, we cannot use this equation to find the smallest values for A and B.