Question

When asked how old she is, a teacher responded with this riddle: “My age is a two digit number. When multiplied by seven, the result is a three digit number. When the digit 6 is written after the three digit number, that number is increased by 1833.” How old is the teacher?

Answer 1

Let x be the age of the teacher.

We know that 7x is a three digit number.

Writing 6 after this number means to multiply this number by 10 (this adds a zero after the number) and then add 6.

So, we actually only need the last information:

$\underbrace{70x+6}_{\text{10 times the 3 digits number plus 6}} = \underbrace{7x+1833}_{\text{the 3 digits number increased by 1833}}$

Solving this equation is quite trivial: subtract 7x  and 6 from both sides and you get

$63x = 1827 \iff x = 29$

This is indeed a two digit number, and if you multiply it by 6 it becomes 174, a three digits number.