Number Puzzle Based on Multiplication IdeasDate: 08/07/2005 at 11:29:58 From: Kate Subject: Find the smallest natural number n...... Find the smallest natural number n which has the following properties: * Its decimal representation has 6 as the last digit * If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is 4 times as large as the original number n. I know that the natural number n has ...46 as last 2 digits. When the digit 6 is shifted to front, then it will be 6...4. Also, before shifting the digit 6, the most front digit must be less than 6. That's what I have realized and tested, but I still can't work it out. Date: 08/08/2005 at 00:15:29 From: Doctor Greenie Subject: Re: Find the smallest natural number n...... Hi, Kate -- You are off to a great start when you noted that the last 2 digits of the number must be "46". How did you determine that? Presumably, you thought something like this: ???????6 x 4 -------- ???????4 Then, according to the statement of the problem, the next-to-last digit in the number must be "4". So now you had ??????46 x 4 -------- ???????4 But we can continue using this same reasoning over and over until we find a solution to the problem. Because we know the last two digits of the original number are "46", we can perform the multiplication by 4 of these two final digits to determine that the second digit from the right in the product is "8": ??????46 x 4 -------- ??????84 Now we know the last two digits of the product; and according to the rules of the problem, the "8" is the third digit from the right in the original number. Then, by performing the multiplication by 4 of the digits we now know of the original number, the third digit from the right in the product is uniquely determined: ?????846 x 4 -------- ?????384 And we just continue this process--multiplying the last digit we found in the original number by 4 to find the next digit to the left in the product, and copying that digit as the next digit to the left in the original number: ????3846 x 4 -------- ????5384 ???53846 x 4 -------- ???15384 ??153846 x 4 -------- ??615384 We now have a "6" as the first digit of the product; this means we are done. The original number is 153846; multiplied by 4 the product is 615384. We can also work the problem in the opposite direction by a similar process. You noted in your original message that the first digit of the original number must be "less than 6". In fact, if the original number multiplied by 4 is to have "6" as the first digit, then the first digit of the original number must be "1". So we have 1???????? x 4 --------- 6???????? But now, according to the rules for the problem, the "1" which is the first digit of the original number must be the second digit of the product: 1???????? x 4 --------- 61??????? Now, by dividing the portion of the product we know by 4, we can see 61/4 = 15 (plus a remainder) so the second digit of the original number must be "5"--and that means the third digit of the product is "5": 15??????? x 4 --------- 615?????? Again, we can repeat this process until we find the solution to the problem: 615/4 = 153 (plus a remainder) so next digit is "3": 153?????? x 4 --------- 6153????? 6153/4 = 1538 (plus a remainder) so next digit is "8": 1538????? x 4 --------- 61538???? 61538/4 = 15384 (plus a remainder) so next digit is "4": 15384???? x 4 --------- 615384??? 615384/4 = 153846 (with no remainder) Since there is no remainder, we are done. Again we have (of course) found the same answer: the original number is 153846, and the product when multiplied by 4 is 615384. Please write back if you have questions on any of this. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/ Date: 08/08/2005 at 06:05:07 From: Kate Subject: Thank you (Find the smallest natural number n......) Doctor Greenie, Thanks for your help in solving this question. I appreciate it, really! This is really a good place to learn math! I love it! |
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