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Logic Level 3

Find the smallest natural number which has the following properties:

(a) Its decimal representation has 6 as the last digit.

(b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number.


The answer is 153846.

Arko Roychoudhury by arko roychoudhury , 20, India

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1 solution

Alex Zhong
Apr 14, 2015

Let A A be the remaining digits such that

4 ( 10 A + 6 ) = 6 × 1 0 k + A ⟹ A = 6 × 1 0 k − 24 39 , 4(10A + 6) = 6\times 10^k +A \implies A=\dfrac{6\times10^k -24 }{39},

where k k is the total number of digits of n n .

Next we do case works from k = 1 , 2 , 3 , … k=1,2,3,\ldots and find that when k = 5 , k=5, A = 15384 A=15384 is an integer. Therefore, n = 153846 . n=\boxed{153846}.

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This is my attempt to this question! Please go through this! We know last digit is 6! When we multiply 6 by 4, we get 4 as last digit. So, second last digit should be four. Also first digit should be 1(If the first digit is >1 and when the number is multiplied by 4, we get the first digit not =6). So, we know first, second last digit and last digit, which are - 1,4,6 respectively. Now, when this no is multiplied by 4, second digit is last digit of-(4 4 +2(carry from 6)) which is 8. So, third last digit is 8. We perform this step until we get 1 as last digit(which is first digit of required number). Therefore, fourth number from back is last digit of (8 4 + 1(carry from 8)) which is 3. Now, the no is 1...3846. Similarly, fifth number from back is last digit of (3 4 +3(carry from 8)) which is 5. Now the no is 1...53846. Now, again finding previous no. It is last digit of (5 4+1(carry from 5)) which is 1! WE GOT 1 !!! Therefore, required number is : 153846.

Rishav Agarwal - 6 years, 2 months ago

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Yeah, I had the same method. The solution above was just the mathematical formalization of what we did.

Swapnil Das - 5 years, 6 months ago

This is an excellent solution!!

Aran Pasupathy - 6 years, 1 month ago

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Thank You!

Rishav Agarwal - 6 years, 1 month ago

I would like to have 153846 dollars. But 6153846 is four times better.

Hasmik Garyaka - 2 years, 6 months ago

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