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

• To determine the digit P, we use the divisibility rule for 7, which involves calculating the alternating sum of the number's digits from right to left (or using the 1, -1, 1, -1... pattern on blocks of three digits, but the digit-wise alternating sum is a valid shortcut for this specific problem structure often used in CSAT).

• Let's apply the standard divisibility test for 7: Calculate the difference between the sum of digits in odd positions and the sum of digits in even positions (counting from the right, position 1 is the units digit). Note: A more robust method is the '1, -1, 1, -1' alternating sum of individual digits, which is equivalent to checking divisibility by 11, but for 7, we must use the specific 7-divisibility algorithm or modular arithmetic.

• Let's use modular arithmetic: $3798125P369 \equiv 0 \pmod 7$.

• We can break the number into parts or use the property that $10 \equiv 3 \pmod 7$, $100 \equiv 2 \pmod 7$, $1000 \equiv 6 \equiv -1 \pmod 7$.

• Let's compute the value of the number modulo 7 by processing digits from right to left with powers of 10:

• $9 \times 10^0 = 9 \equiv 2$

• $6 \times 10^1 = 60 \equiv 4$

• $3 \times 10^2 = 300 \equiv 6$

• $P \times 10^3 = 1000P \equiv -P$

• $5 \times 10^4 = 50000 \equiv 5 \times 4 = 20 \equiv 6$

• $2 \times 10^5 = 200000 \equiv 2 \times 6 = 12 \equiv 5$

• $1 \times 10^6 = 1000000 \equiv 1 \times 2 = 2$

• $8 \times 10^7 = 80000000 \equiv 8 \times 3 = 24 \equiv 3$

• $9 \times 10^8 = 900000000 \equiv 9 \times 5 = 45 \equiv 3$

• $7 \times 10^9 = 7000000000 \equiv 7 \times 6 = 42 \equiv 0$

• $3 \times 10^{10} = 30000000000 \equiv 3 \times 4 = 12 \equiv 5$

• Summing these residues: $2 + 4 + 6 - P + 6 + 5 + 2 + 3 + 3 + 0 + 5 = 36 - P$.

• For the number to be divisible by 7, $36 - P \equiv 0 \pmod 7$.

• $36 \equiv 1 \pmod 7$ (since $35$ is divisible by 7). So, $1 - P \equiv 0 \pmod 7$, which implies $P = 1$ or $P = 8$.

• Wait, let me re-verify the powers of 10 mod 7:

• $10^0 = 1$

• $10^1 = 3$

• $10^2 = 2$

• $10^3 = 6 (-1)$

• $10^4 = 4$

• $10^5 = 5 (-2)$

• $10^6 = 1$

• $10^7 = 3$

• $10^8 = 2$

• $10^9 = 6 (-1)$

• $10^{10} = 4$

• Recalculating with correct powers:

• $9 \times 1 = 9 \equiv 2$

• $6 \times 3 = 18 \equiv 4$

• $3 \times 2 = 6$

• $P \times 6 = 6P \equiv -P$

• $5 \times 4 = 20 \equiv 6$

• $2 \times 5 = 10 \equiv 3$

• $1 \times 1 = 1$

• $8 \times 3 = 24 \equiv 3$

• $9 \times 2 = 18 \equiv 4$

• $7 \times 6 = 42 \equiv 0$

• $3 \times 4 = 12 \equiv 5$

• Sum: $2 + 4 + 6 - P + 6 + 3 + 1 + 3 + 4 + 0 + 5 = 34 - P$.

• $34 \equiv 6 \pmod 7$ (since $28$ is divisible by 7, $34-28=6$).

• So, $6 - P \equiv 0 \pmod 7 \Rightarrow P = 6$.

• Thus, the digit P is 6.