2006 iTest Problems/Problem 3

Revision as of 02:30, 27 November 2018 by Rockmanex3 (talk | contribs) (Solution to Problem 3 — products and optimization)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Problem

Let $I, T, E, S$ be distinct positive integers such that the product $ITEST  =  2006$. What is the largest possible value of the sum $I  +  T  +  E  +  S  +  T  +  2006$?

$\mathrm{(A)}\, 2086\quad\mathrm{(B)}\, 4012\quad\mathrm{(C)}\, 2144$

Solution

Note that the expression can be written as $IT^2ES = 2006$. The prime factorization of $2006$ is $2 \cdot 17 \cdot 59$. Since $2006$ does not have perfect squares as factors, $T = 1$. The other variables are the three primes since all variables are distinct, so the maximum value of $I  +  T  +  E  +  S  +  T  +  2006$ is $2 + 2 + 17 + 59 + 2006 = \boxed{\textbf{(A) } 2086}$.

See Also

2006 iTest (Problems)
Preceded by:
Problem 2
Followed by:
Problem 4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 U1 U2 U3 U4 U5 U6 U7 U8 U9 U10
Invalid username
Login to AoPS