Challenging mathematics question

Question

An integer has 2018 digits and is divisible by 7. The first digit is d, while all the other digits

are 2. What is the value of d?

Answer 1

take the last digit, multiply by 2, then subtract from the preceding digits, then divide by 7

if it's = 7 or 0, it's divisible by 7

example

112 has 2 at the end. 2x2 =4

11-4 = 7 which is divisible by 7 with remainder 0, so 112 is divisible by 7

the general method for finding a number divisible by 7

take the unit digit, multiply by 2, subtract from the number formed by the preceding digits and divide by 7.

if it divides evenly then the original number is divisible by 7

7 divides 42,222,222 evenly.

7 divides 4 followed by 6 two's followed by one more two

or any number of sets of 6 two's. 216/6 =36

218 digits with a 4 in front have 217 twos. 217/6 = 36 with remainder of 1

d=4 works