Divisibility Rules Level - 2

A 6-digit number is formed by repeating a 3-digit natural number (e.g., 256256). What is the largest number that always divides such a number?

Find the remainder when the 100-digit number 7777...77 is divided by 101.

A 96-digit number N is formed by writing the digit 7 ninety-six times. Another number M is formed by writing 3 ninety-six times. What is the remainder when N x M is divided by 1001?

What is the remainder when N = 10³ + 10⁶ + 10⁹ + ... + 10⁹⁹ is divided by 999?

If the 8-digit number 34A56B23 is divisible by 99, find the value of A+B.

How many 4-digit numbers of the form ABBA are divisible by 101?

A number N, when split into blocks of 3 digits from right to left and summed, gives a multiple of 37. Which must N always be divisible by?

Find the remainder when 10¹²- 1 is divided by 1001.

Let X = 555...55 (60 digits) and Y = 999...99 (60 digits). Which of the following primes does not divide X x Y?

Let A be a 50-digit repunit (111...11). Find the remainder when C = A x (10⁵⁰ +1) + 10¹⁰⁰  is divided by 101.