Application of Divisibility Rules
Prove the divisibility rule of 7: For a number double the units digit and take its difference with the remaining number except for the units digit. If it is a multiple of 7, then the number is divisible by 7.
This follows because , so is a multiple of 7 if and only if is a multiple of 7. If we use , we get the result as stated.
Show that if the last 3 digits of a number are , then is a multiple of 8 if and only if is a multiple of 8.
This follows because .
Hence, by the divisibility rule of 8, is a multiple of 8 if and only if is a multiple of 8.
Show that the 6-digit number is a multiple of 7 if and only if is a multiple of 7.
This follows because
Hence is a multiple of 7 if and only if is a multiple of 7.