Replace "stars" with different natural numbers in order to obtain an identity:
\[\frac{1}{*}+\frac{1}{*}+\frac{1}{*}=\frac{1}{*}+\frac{1}{*}+\frac{1}{*};\]
George knows a representation of number “8” as the sum of its divisors in which only divisor “1” appears twice: \[8=4+2+1+1.\] His brother showed George that such representation exists for number “16” as well: \[16=8+4+2+1+1.\] He apologies for forgetting an example considering number “32” but he is sure once he saw such representation for this number.
(a) Help George to work out a suitable representation for number “32”;
(b) Can you think of a number which has such representation consisting of 7 terms?
(c) Of 11 terms?
(d) Can you find a number which can be represented as a sum of its divisors which are all different (pay attention that we don’t allow repeating digit “1” twice!)?
(e) What if we require this representation to consist of 11 terms?
George claims that he knows two numbers such that their quotient is equal to their product. Can we believe him? Prove him wrong or provide a suitable example.
In the context of Example 14.2 what is the answer if we have five numbers instead of four? (i.e., can we get four distinct prime numbers then?)
Now George is sure he found two numbers with the quotient equal to their sum. And on top of that their product is still equal to the same value. Can it be true?
Find the biggest 6-digit number such that each of its digits, except for the last two, is equal to the sum of its two right neighbours.
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 fit 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 first two, is equal to the sum of its two left neighbours. (Compared to part (a), we removed the “6-digit number” restriction.)
A six-digit number starts with the digit \(1\). If this digit is relocated to the rightmost position, the number becomes \(3\) times bigger. What is the number?
Shmerlin managed to enter the cave and explore it. On his way back, he was once again stopped by Drago. He learns that the door out of the cave is locked again, this time with a more powerful lock. The key required to open it now includes four positive integers, which are no longer digits – they can be much larger. Shmerlin once again can choose four integer numbers: \(x, y, z\) and \(w\), and the dragon will tell him the value of \(A \times x + B \times y + C \times z + D \times w\), where \(A, B, C\) and \(D\) are the four secret integer numbers that open the lock. Because the lock is much more difficult to crack now, Drago agrees to let Shmerlin try twice. He can choose his four integer numbers and then, basing on what he learns from the dragon, choose again. Will he be able to leave the cave or is he doomed to stay inside forever?