IMC 2013 Q25
25. In 1984 the engineer and prolific prime-finder Harvey Dubner found the biggest known prime each of whose digits is either a one or a zero. The prime can be expressed as
\(\dfrac{10^{641} \times (10^{640} - 1)}{9} + 1\). How many digits does this prime have?
- A 640
- B 641
- C 1280
- D 1281
- E 640 × 641
Intermediate Mathematical Challenge 2013, Question 25. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 1281
\((10^{640} - 1)\) is a 640-digit number consisting entirely of nines. So \(\dfrac{(10^{640} - 1)}{9}\) is a 640-digit number consisting entirely of ones.
Therefore \(\dfrac{10^{641} \times (10^{640} - 1)}{9}\) consists of 640 ones followed by 641 zeros. So \(\dfrac{10^{641} \times (10^{640} - 1)}{9} + 1\) consists of 640 ones followed by 640 zeros followed by a single one. Therefore it has 1281 digits.
Solution © UK Mathematics Trust, from the IMC 2013 solutions. Longer versions with investigations: Solutions and investigations (PDF).