Multiply to get simply 1's and 0's
Algebra
Level
5
All integers have multiples having 1's and 0's, e.g. 7 x 143 = 1001. What is the smallest multiple of 7827 such that the product is having 1's and 0's only? Note: A 10-digit calculator may be of help.
The answer is 127763.
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1 solution
As 7827 is a multiple of 3, any multiple thereof must have sum of digits also a multiple of 3. Hence we first look for 1's and 0's sequence with exactly three 1's. Unit's digit has to be 1 in the case of smallest multiple. Now the remainders are 10, 100, 1000, 2173, 6076, 5971, 4921, 2248, 6826, ......etc. for 10's of thousands. It is easy to see that 1 + 1000 + 6826 = 7827. Hence 1000001001 is a multiple of 7827. On division we get answer as 127763.