The difference of 10^{31}-5 and 10^{30}+p is divisible by 3 where p is a digit. How many values of p are possible?

Explanation

Using modulo 3, 10 \equiv 1 \pmod 3. The expression (10^{31}-5) - (10^{30}+p) \equiv 1 - 5 - 1 - p \equiv -5 - p \equiv 1 - p \pmod 3. For divisibility by 3, p \equiv 1 \pmod 3. Single-digit values for p are 1, 4, and 7, giving 3 possibilities.

Related questions on Arithmetic

• What is the remainder when (17^{25} + 19^{25}) is divided by 18?
• A bottle contains spirit and water in the ratio 1:4 and another identical bottle contains spirit and water in the ratio 4:1. In what rat...
• Let P = 5^5 \times 15^{15} \times 25^{25} \times 35^{35} and Q = 10^{10} \times 20^{20} \times 30^{30} \times 40^{40}. What is the numbe...
• Two students X and Y appeared in a test. The score of X is 20 more than that of Y. If the score of X is 75% of the sum of the scores of X an...
• Question: The product of a natural number N and the number M written by the same digits of N in the reverse order is 252. What is the number...