The number 3798125P369 is divisible by 7. What is the value of the digit P?

To determine the value of the digit P such that the number 3798125P369 is divisible by 7, we use the divisibility rule for 7 based on alternating sums of blocks of three digits from the right. 1. **Identify the blocks of three digits from the right:** * Block 1: 369 (last three digits) * Block 2: 25P (the next three digits, where P is the units digit of this block) * Block 3: 981 (the next three digits) * Block 4: 37 (the remaining two digits) 2. **Apply the alternating sum rule:** The number is divisible by 7 if the alternating sum of these blocks is divisible by 7. Sum = (Block 1) - (Block 2) + (Block 3) - (Block 4) Sum = 369 - (250 + P) + 981 - 37 3. **Simplify the expression:** Sum = 369 - 250 - P + 981 - 37 Sum = (369 + 981) - (250 + 37) - P Sum = 1350 - 287 - P Sum = 1063 - P 4. **Find P such that (1063 - P) is divisible by 7:** First, find the remainder of 1063 when divided by 7: 1063 = 7 * 151 + 6 So, 1063 is congruent to 6 (mod 7). Therefore, we need (6 - P) to be divisible by 7. 5. **Test the given options for P:** * A) P = 1: (6 - 1) = 5. (Not divisible by 7) * B) P = 6: (6 - 6) = 0. (Divisible by 7) * C) P = 7: (6 - 7) = -1. (Not divisible by 7) * D) P = 9: (6 - 9) = -3. (Not divisible by 7) The only value of P that makes (6 - P) divisible by 7 is P = 6. The final answer is B) 6.