Question

Let S be the set of all 4 -digit base 10 palindromes, and let T= a₁,a₂,a₃,...a₁₀ be a 10 -element subset of S. What is the maximum possible value of gcd (a₁,a₂,a₃,...,a₁₀) ?

(A) 55
(B) 66
(C) 88
(D) 99
(E) 132

Answer 1

The maximum possible value of gcd(a₁, a₂, a₃, ..., a₁₀) is 11.

Step-by-step explanation:

To find the maximum possible value of gcd(a₁, a₂, a₃, ..., a₁₀), we need to consider the properties of palindromic numbers. A palindromic number reads the same forwards and backwards. In a 4-digit base 10 palindrome, the first and fourth digits are the same, and the second and third digits are the same. For example, 1221 and 4554 are palindromes.

Since the greatest common divisor (gcd) is the largest number that divides all the given numbers evenly, we need to find a set of palindromes where the smallest common factor among the numbers is as large as possible.

By examining the set S, where S is the set of all 4-digit base 10 palindromes, we can see that the palindromes with a common factor of 11 have the largest gcd. For example, the palindromes 1221, 1331, 1441, and so on, all have a gcd of 11 because 11 is the smallest common factor among them. Therefore, the maximum possible value of gcd(a₁, a₂, a₃, ..., a₁₀) is 11.