Final answer:
To find the sum of the digits of the two-digit number N, we can set up equations using the given remainders. After solving the equations, we find that N = 13. Therefore, the sum of the digits of N is 4. Therefore, the sum of the digits of N is 1 + 3 = 4.
Step-by-step explanation:
To find the sum of the digits of the two-digit number N, we need to solve the equations:
272,758 / N = Q1 + R1
273,437 / N = Q2 + R2
Where Q1 and Q2 are the quotients, and R1 and R2 are the remainders.
Since the remainders are given as 13 and 17, we can set up the following equations:
272,758 = N * Q1 + 13
273,437 = N * Q2 + 17
We can subtract the two equations to eliminate the N:
273,437 - 272,758 = N * (Q2 - Q1) + (17 - 13)
679 = N * (Q2 - Q1) + 4
Now, we know that N is a two-digit number, so it must be between 10 and 99.
We can try different values for N and solve for (Q2 - Q1) until we find a value of N that satisfies the equation.
Let's start with N = 11:
679 = 11 * (Q2 - Q1) + 4
675 = 11 * (Q2 - Q1)
61 = Q2 - Q1
The difference between two-digit numbers can't be less than 10, so N = 11 doesn't work.
Let's try N = 12:
679 = 12 * (Q2 - Q1) + 4
671 = 12 * (Q2 - Q1)
55 = Q2 - Q1
The difference between two-digit numbers can't be 0, so N = 12 doesn't work either.
Continuing this process, we can try different values of N until we find a value that satisfies the equation.
After trying all possible values for N, we find that N = 13 is the only value that satisfies the equation.
Therefore, the sum of the digits of N is 1 + 3 = 4.