Problems

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Jane wrote a number on the whiteboard. Then, she looked at it and she noticed it lacks her favourite digit: 5. So she wrote 5 at the end of it. She then realized the new number is larger than the original one by exactly 1661. What is the number written on the board?

Replace letters with digits to maximize the expression: \[NO + MORE + MATH\] (same letters stand for identical digits and different letters stand for different digits.)

Jane wrote another number on the board. This time it was a two-digit number and again it did not include digit 5. Jane then decided to include it, but the number was written too close to the edge, so she decided to t the 5 in between the two digits. She noticed that the resulting number is 11 times larger than the original. What is the sum of digits of the new number?

a) Find the biggest 6-digit integer number such that each digit, except for the two on the left, is equal to the sum of its two left neighbours.

b) Find the biggest integer number such that each digit, except for the rst two, is equal to the sum of its two left neighbours. (Compared to part (a), we removed the 6-digit number restriction.)

Prove the divisibility rule for \(3\): the number is divisible by \(3\) if and only if the sum of its digits is divisible by \(3\).

While studying numbers and its properites, Robinson came across a 3-digit prime number with the last digit being equal to the sum of the first two digits. What was the last digit of that number if among the number did not have any zeros among it’s digits?

Prove the divisibility rule for \(4\): a number is divisible by \(4\) if and only if the number made by the last two digits of the original number is divisible by \(4\);
Can you come up with a divisibility rule for \(8\)?

Robinson found a chest with books and instruments after the ship wreck. Not all the books were in readable condition, but some of the books he managed to read. One sentence read “72 chickens cost *619* p”. (The starred digits were not readable). He has not tasted a chicken for quite some time, and it was pleasant to imagine a properly cooked chicken in front of him. He also was able to decipher the cost of one chicken. Can you?

When Robinson Crusoe’s friend and assistant named Friday learned about divisibility rules, he was so impressed that he proposed his own rule:

a number is divisible by 27 if the sum of it’s digits is divisible by 27.

Was he right?