Logical Number Theory

Number Theory Level 4

Each alphabet is represented as a number as shown in the image above. So now you choose an arbitrary word and represent it in the form of numbers above and let $P$ be the product of these numbers.

Find the smallest three digit composite number which cannot take the value of $P$ for any arbitrary word.

Details And assumptions:

You can choose any word (whether it makes sense or not does not matter).
As an explicit example , if the word is $nihar$ , then $P=14\times 9 \times 8 \times 1 \times 18 = 18144$ .
This problem is original.

The answer is 106.

2 solutions

Nihar Mahajan
Jun 22, 2015

We can have product of numbers from $1$ to $26$ .Note that if $P$ contains a prime which is greater than $26$ , then it cannot represent any word. So lets start with $101$ .

$101= \ prime \\ 102 = 17 \times 2 \times 3 = \ possible \\ 103 = \ prime \\ 104 = 8\times 13 = \ possible \\ 105 = 3\times 5 \times 7 = \ possible \\ 106 = 2 \times 53 = \ not \ possible \ since \ 53>26$

Hence $106$ is the answer.

i did the same but is it mathematical valid?

Akash singh - 5 years, 9 months ago

101>26, therefore no combination of letters. I don't understand your arbitrary rule forcing 106.

Erica LaVoi - 5 years, 11 months ago

Please read the rules properly , it states that we are in need of the smallest three digit "COMPOSITE" number and not prime.

Nihar Mahajan - 5 years, 11 months ago

Mukul Sharma
Jul 7, 2015

ITS SIMPLE; we just got to find a 3 digit composite number whose prime factorization involves atleast one prime no. that does'nt come between 1-26

Logical Number Theory

The answer is 106.

2 solutions

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