# Problems of the Week

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# 2018-11-12 Basic

When will it pop?

Six dark-grey symbols (L, H, U, S, T, $$+$$) are drawn on six identical pieces of square paper.

How many of these symbols have the same perimeter as the square paper itself?

You are given two integers $$J$$ and $$K.$$

At most, how many of the following statements can be simultaneously true?

• $$J + K$$ is an odd number.
• $$J - K$$ is an odd number.
• $$J \times K$$ is an odd number.
• $$J \div K$$ is an odd number.

I am looking for a positive integer greater than 9 such that the product of all of its digits is equal to the integer itself:

$\underbrace{{\color{red}a} \times {\color{blue}b} \times {\color{green}c} \times \cdots \times {\color{indigo}x} \times {\color{purple}y} \times {\color{pink}z} = \overline{{\color{red}a}{\color{blue}b}{\color{green}c} \ldots {\color{indigo}x}{\color{purple}y}{\color{pink}z}}}_{\text{The product of the digits is equal to the integer itself.}}.$

Are there any positive integers that satisfy this condition?

$\sqrt1,\ \sqrt{11},\ \sqrt{111},\ \sqrt{1111},\ \sqrt{11111},\ \ldots$

Is it true that $$\sqrt{1}$$ is the only integer in this list of numbers?

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