Problems

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Found: 42

Does there exist a natural number which, when divided by the sum of its digits, gives a quotient and remainder both equal to the number 2011?

Prove that: a1a2a3an1an×103an1an×103(mod4), where n is a natural number and ai for i=1,2,,n are the digits of some number.

How many different four-digit numbers, divisible by 4, can be made up of the digits 1, 2, 3 and 4,

a) if each number can occur only once?

b) if each number can occur several times?

It is known that 35!=1033314796638614492966651337523200000000. Find the number replaced by an asterisk.

Prove the divisibility rule for 25: a number is divisible by 25 if and only if the number made by the last two digits of the original number is divisible by 25;
Can you come up with a divisibility rule for 125?