1.
a) Add one digit to the left and right of the number 15 so that the resulting number is divisible by 15.
b) Add one digit to the left and right of the number 10 so that you get a number that is a multiple of 72.
2. A certain number is divisible by 4 and 6. Does it have to be divisible by 24?
3. Find the largest natural number divisible by 36, the notation of which involves all 10 digits once.
4. On the board it is written: 645*7235. Replace the asterisk with a number so that the resulting number is divisible by 3.
5. Replace the asterisks in the number 72*3* with digits so that the number is divisible by 45 without a remainder.
6. In the country of Anchuria, the following denominations of banknotes are in circulation: 1 anchur, 10 anchur, 100 anchur, 1000 anchur. Is it possible to count out a million anchurs so that you get exactly half a million banknotes?
7. a) Find a two-digit number whose first digit is equal to the difference between this number and a number written with the same digits, but in reverse order.
b) Solve the puzzle AB – BA = A.
8. Is it true that if you write the digits of any integer in reverse order, then the difference between the original and new numbers will be divisible by 9?
9. Find all two-digit numbers whose sum of digits does not change after multiplication by either 2, 3, 4, ..., 8, or 9.
1Is there a natural number whose product of digits equals 528?
11. How many digits are in the number 11...11 if it is divisible by 999,999,999?
12. The numbers were rearranged and a number was obtained that was 3 times smaller than the original one. Prove that the original number is divisible by 27.
13. The sum of its digits was added to the number. The sum of its digits was added to the resulting number, and so on. When the sum of its digits was added to the number for the seventh time, we got 1000. What number did you start with?
14. For each of the numbers from 1 to 1,000,000,000, the sum of its digits was calculated, for each of the resulting billion numbers, the sum of the digits was again calculated, and so on until we got a billion single-digit numbers. What numbers did you get the most?
15.
a) Prove that numbers of the form aa, abcabc, abcdeabcde are divisible by 11. (And in general, prove that if we assign the same number to an arbitrary number that has an odd number of digits, we get a number divisible by 11.)
b) If you add to an arbitrary number a number written with the same digits in reverse order, then the resulting number will be divided by 11 without a remainder: for example, numbers of the form aa, abba, abccba are multiples of 11. Prove this.
Be up-to-date with our recent updates, new problems and answers!
