Lesson Progress

0% Complete

The digits , , , , and are each used once to write a five-digit number . The three-digit number is divisible by , the three-digit number is divisible by , and the three-digit number is divisible by . What is ?

Easy

Mathematical Circles

Do you really need a hint? Try it first!

We see that since QRS is divisible by , must equal either or , but it cannot equal , so . Similarly proceed .

We notice that since PQR must be even, R must be either or . However, when \( R = 2 \) we see that \( T \equiv 2 (mod \ 3) \), which cannot happen because and are already used up; so .

This gives meaning . Almost we are done . Try to think the rest .

Now, we see that Q could be either or , but is not divisible by , but is. This means that and the P is clearly equal to 1.

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.