Crunchy Coconuts

Number Theory Level 4

A positive number $n$ is called a coconut if the logarithm of $n$ to the base 10 is in the interval $[2,3)$ .

Moreover, a natural number $n$ is called crunchy if it suffices the following condition $\displaystyle \text{SOD}(3+n)=\dfrac{\text{SOD}(n)}{3}$
,where $\displaystyle \text{SOD}(n)$ is an operator which tells the sum of digits of the number $\displaystyle n$ .

What is the sum of all numbers that are both crunchy and coconuts' ?

The answer is 432.

2 solutions

Karthik Perumal Palaniappan
May 21, 2015

Since the characteristic of the logarithm of n to the base 10 is 2, the number n has to be in between 100 and 999. So it is a 3 digit number.

We represent the 3 digit number as 100 a + 10 b + c where a,b,c are the digits of the number.

Hence SOD(n) = a + b + c

Now, SOD(n + 3) has two cases. If 0 <= c <= 6, then the SOD(n +3) = a + b + c + 3.

If 7 <= c <= 10, then the SOD(n + 3) = a + (b + 1) + (c - 7) = a + b + c - 6. The tens digit is incremented by 1 and the units digit is 0, 1 or 2 depending on whether c is 7, 8 or 9.

Now consider the equation SOD(3 + n) = SOD(n) / 3.

Case 1: 0 <= c <= 6

a + b + c + 3 = (a + b + c)/3 . Solving we get (a + b + c) = -4.5 which is not possible.

Case 2: 7 <= c <= 9

a + b + c - 6 = (a + b + c)/3. Solving we get (a + b + c) = 9.

We know that the value of c can only be 7, 8 or 9. If the value of c is 9, then a and b have to be 0 which means it is not a 3 digit number. If the value is 8, the only possible value of n is 108. If the value is 7, the two possible values of n are 117 and 207.

Adding 108, 117 and 207 we get 432. Hence the answer is 432.

Taj Muhammed
May 19, 2015

If abc is 3 digit number, a at place of hundred, b at place of ten and c at unit's place, then cheracteristic of 2 implies that natural number lie between 100 and 999. Now consider following scenarios: First case is when c is less than or equal to 6, SOD(3+n) cannot equal SOD(n)/3. Second case is when c is greater than 6 but b is less than or equal to 8. Adding 3 to that number and imposing the condition that SOD(new)= SOD(old)/3 implies that a + b + c equals 9. With c between 7 and 9, this yield three numbers i.e. 108, 117 and 207. Thirdly when c is greater than 6 but b equal to 9 and a is less than or equal to 8. Adding 3 to that number and imposing the condition that SOD(new)= SOD(old)/3 implies that a + b + c equals 22.5, which is not possible. Lastly, we can manually check 3 numbers 997, 998 and 999. They don't satisfy our conditions. Hence, sum of 108, 117 and 207 is our desired result.

Crunchy Coconuts

The answer is 432.

2 solutions

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