Let N = 10p78pq (7digit number). If N is exactly divisible by 120 then the sum…

Step-by-Step Solution

To determine the sum of the digits for the 7-digit number N = 10p78pq that is divisible by 120, we must apply divisibility rules.

1. Analyze Divisibility by 120:

   • A number divisible by 120 must be divisible by its factors that are coprime, such as 3, 5, and 8 (since 3 * 5 * 8 = 120).

   • Divisibility by 10: For N to be divisible by 120, it must end in 0. Therefore, q = 0.

   • Now N = 10p78p0.

2. Divisibility by 8:

   • A number is divisible by 8 if its last three digits are divisible by 8.

   • The last three digits are 8p0.

   • Testing values for p:

     • If p = 0: 800 / 8 = 100 (Works)

     • If p = 4: 840 / 8 = 105 (Works)

     • If p = 8: 880 / 8 = 110 (Works)

3. Divisibility by 3:

   • A number is divisible by 3 if the sum of its digits is divisible by 3.

   • Sum = 1 + 0 + p + 7 + 8 + p + 0 = 16 + 2p.

   • Test the possible values of p:

     • If p = 0: Sum = 16 (Not divisible by 3)

     • If p = 4: Sum = 16 + 8 = 24 (Divisible by 3)

     • If p = 8: Sum = 16 + 16 = 32 (Not divisible by 3)

4. Calculate Final Sum:

   • With p = 4 and q = 0, the number N = 1047840.

   • Sum of the digits = 1 + 0 + 4 + 7 + 8 + 4 + 0 = 24.