Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Basics of Mathematics

Olympiad workbook · IOQM · Basics of Mathematics

Let N be the smallest positive integer such that N+2 ~N+3 ~N+ +9 ~N...

Let N be the smallest positive integer such that N+2 N+3 N+…+9 N\mathrm{N}+2 \mathrm{~N}+3 \mathrm{~N}+\ldots+9 \mathrm{~N} is a number all whose digits are equal. What is the sum of the digits of NN ?

Correct answer

37

Step-by-step solution

N∈Z+P=N+2 N+3 N+………+9 N=45 N45=555………..5⏟a times \begin{aligned} & \mathrm{N} \in \mathrm{Z}^{+} \\ & \mathrm{P}=\mathrm{N}+2 \mathrm{~N}+3 \mathrm{~N}+\ldots \ldots \ldots+9 \mathrm{~N}=45 \mathrm{~N} \\ & 45=\underbrace{555 \ldots \ldots \ldots . .5}_{\text {a times }} \end{aligned} Only 55……….555 \ldots \ldots \ldots .5 can be repeated as 45 N will have units place =0=0 or 5 Also as 45 N→45 \mathrm{~N} \rightarrow multiple of ' 9 ' also. ∴ If a=9⇒55555555545=N∴N=12345679\begin{aligned} & \therefore \text { If } a=9 \quad \Rightarrow \frac{555555555}{45}=N \\ & \therefore N=12345679 \end{aligned} Sum of digits of N=37\mathrm{N}=37.
Practice Basics of Mathematics on Quantrex Academy →

More from Basics of Mathematics

All Basics of Mathematics questions Full Basics of Mathematics list All Olympiad workbook PYQs