Guest Posted September 6, 2009 Report Share Posted September 6, 2009 (edited) Y is a base ten positive integer between 1 and 100 inclusively, such that 10Y is expressible as the product of two positive integers, neither of which contains the digit zero. Determine the maximum value of Y for which this is possible. Edited September 6, 2009 by K Sengupta Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted September 6, 2009 Report Share Posted September 6, 2009 Minimum is easier. 101 = 2x5 OK now I'll work on the maximum. Quote Link to comment Share on other sites More sharing options...
0 superprismatic Posted September 6, 2009 Report Share Posted September 6, 2009 Y is a base ten positive integer between 1 and 100 inclusively, such that 10Y is expressible as the product of two positive integers, neither of which contains the digit zero. Determine the maximum value of Y for which this is possible. 33 Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted September 6, 2009 Report Share Posted September 6, 2009 Let 100n = 2n x 5n. Checking the first 16 digits of 2n and 5n permits Y=18. Values from 19 to 32, 34, 35, and 38-40 have zeros in first 16 digits. Y=33, 36 and 37 do not, but I can't check their remaining digits. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted September 6, 2009 Report Share Posted September 6, 2009 the correct answer for max value I believe is: 64...next is 128 but that is more than 100 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted September 6, 2009 Report Share Posted September 6, 2009 33 are you sure about 33....what are the two numbers that I am missing that when multiplied together gives 10^x where x is 33 Quote Link to comment Share on other sites More sharing options...
0 superprismatic Posted September 6, 2009 Report Share Posted September 6, 2009 are you sure about 33....what are the two numbers that I am missing that when multiplied together gives 10^x where x is 33 8589934592 * 116415321826934814453125 = 1000000000000000000000000000000000 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted September 7, 2009 Report Share Posted September 7, 2009 (edited) KS specified 1 <= Y <= 100 in his question. I can't prove it, but I doubt that there is any other number greater than that mentioned by the solvers which satisfies the "no zeroes" condition. I have tested up to Y <= 10,000. Edited September 7, 2009 by jerbil Quote Link to comment Share on other sites More sharing options...
0 superprismatic Posted September 7, 2009 Report Share Posted September 7, 2009 KS specified 1 <= Y <= 100 in his question. I can't prove it, but I doubt that there is any other number greater than that mentioned by the solvers which satisfies the "no zeroes" condition. I have tested up to Y <= 10,000. I also tested more: Y <= 1,000,000 and found no more. The highest power of 5 I found with no 0s in the base ten representation was 558=34694469519536141888238489627838134765625 and the highest such power of 2 was 286=77371252455336267181195264 which means that their product is 268435456 followed by 58 zeros! Quote Link to comment Share on other sites More sharing options...
0 Guest Posted September 9, 2009 Report Share Posted September 9, 2009 10^8 = 2^8 X 5^8 = 256 X 390625 The maximum value for Y is 7. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted September 9, 2009 Report Share Posted September 9, 2009 10^8 = 2^8 X 5^8 = 256 X 390625 The maximum value for Y is 7. Er ...... No! Quote Link to comment Share on other sites More sharing options...
0 Guest Posted September 16, 2009 Report Share Posted September 16, 2009 (edited) The trick to this problem is to realize that the prime factorization of 10^Y for any Y is 2^Y * 5^Y. As such all factors , prime or otherwise, are expressed as combinations of one or more 2s and 5s. Note that 1 and 10^Y are also non-prime factors (depending upon if you consider 1 prime or not), but clearly the pair will not work as a solution to the problem. Now the problem says that neither of the the pair of factors chosen can contain a zero. Any combination of multiplications between 2 and 5 yield an end zero. For example 2*2*2*5 = 40 or 5*5*5*2 = 250. I am not capable of producing a proof; maybe someone else can help. Therefore, candidate factor pairs must be of the form (2^Y) and (5^Y). This reduces the problem to checking at most 100 factor pairs. From there, I employed the brute force method with some help from a large number calculator, I found on line. Starting from 100 downward, the first instance of both 2^Y and 5^Y in which neither had any zeros occurred at Y=33. Hence, assuming the large number calculator is accurate, Y=33 is the answer. Edited September 16, 2009 by BaltimoreMike Quote Link to comment Share on other sites More sharing options...
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Y is a base ten positive integer between 1 and 100 inclusively, such that 10Y is expressible as the product of two positive integers, neither of which contains the digit zero.
Determine the maximum value of Y for which this is possible.
Edited by K SenguptaLink to comment
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