Find-the-smallest-number-greater-than-zero-which-can-be-written-with-ones-and-zeroes-and-is-evenly-divisble-by-225-

Question Number 515 by 112358 last updated on 25/Jan/15

F i n d t h e s m a l l e s t n u m b e r g r e a t e r

t h a n z e r o w h i c h c a n b e w r i t t e n

w i t h o n e s a n d z e r o e s a n d i s e v e n l y d i v i s b l e

b y 225 .

Answered by prakash jain last updated on 22/Jan/15

225 = 25 \times 9

For 25, 10^{2}

number is of form n = \underset{i = 1}{\sum^{j}} 10^{k_{i}}, k_{i} \in N \cup {0}

n is divisible 9 .

Let say say k_{i} > k_{i - 1} .

It is clear that for smallest n k_{1} = 0

\frac{10^{k_{j}} + 10^{k_{j - 1}} + 10^{k_{1}}}{9} = \frac{(10^{k_{j}} - 1) + (10^{k_{j - 1}} - 1) + \dots + (1 + j - 1)}{9}

RHS is divisible by 9 only for j = 9

So 9 terms are needed in the sum .

For smallest value for n

n = \underset{k = 0}{\sum^{8}} 10^{k} = 111111111

Smallest n of the given form for 225 .

11111111100