Daniel's polynomial palindrome

the equation can be written as (9n+5)(n^2-2)

as this no. is a palindrome , it is divisible by 11

a quick check gives 8 as the value.. 9(8) +5 = 77 , a no. div by 11

putting the value, the required no. is 4774.. sum of digits is 22

our first approach towards the problem is to factorize it which gives (n^2 -2)(9n+5). Now every four digit palindrome is divisible by 11, hence either n^2-2 =11k or 9n+5 = 11k. But for n^2-2=11k, we do not get integer values for 'n'. Hence 9n+5 =11k. Now, for k=7, we get n=8. Which on substituting in the polynomial gives the palindrome as 4774. Hence 4+7+7+4=22.

9n^3+5n^2-18n-10=(9n+5)(n^2-2)=k If n lies between 5 and 10(both inclusive) then only k is a 4 digit number. Check for 5 values which are 5,6,7,8,9 and 10 of n. At n = 8 k is a palindrome which is 4774. 4+7+7+4=22(answer)

Since $9n^3+5n^2-18n-10$ is a 4 digit number and since $n$ is a positive integer, we can easily infer that $5\leq n \leq 10$ . Substituting $n=5,6,7...$ in the polynomial, we see that it is a palindrome only for $n=8$ and the palindrome is $4774$ . Thus, the sum of digits of the palindrome= $4+7+7+4= \boxed{22}$

Can somebody help me find solutions to this question?

Anagram Cracker!!

Anagrams are problems related to shuffled letters which are needed to be arranged and made into perfect meaningful sentences without repeating the letters (letters can be used only once).

Here are some anagrams which you need to crack:

1) tuteauaewribeifslh

2) geaperioitrdspawsagnhabineod

3) enaednenetorfyimrw

Remember to arrange and make a meaningful sentence (one sentence from each group of letters), not single word. If you are able to solve this anagrams please inform me the answers as well as how you found the solutions to the anagrams.

Details and assumptions:

Example:

"My name is Anil" can be written in the form of group of letters as:

meailaysmnni

• Since its a four digit number the possibility of n will be less than 20 .

• But coeff of n^3 is 9 hence n value cant be more than 10 and it cant be 10 as the number will end with 0 incase of 10.

• lets try below 10 in order,incase of 9 we will get 6794.But in case of 8 we are getting 4774.

• Hence the answer will be 22.

n=8

9n^{3} + 5n^{2} - 18n -10 can be factorised as (9n + 5)(n^{2} - 2). Now since the number is a palindrome of 4 digits thus the one of the two components of the factorisation should be a multiple of 11. Now since the number is a 4 digit number, thus the lower and upper bounds of the value of n are 5 and 10 respectively(can be easily checked that anything beyond the range wouldn't be conforming to the 4 digit constraint given in the problem). Now the only value of n for which one of the values of the number is a multiple of 11 is n=8 and the number being 4774 and the answer being 4+7+7+4=22