In this paper, we study the inapproximability of the following NP-complete number theoretic optimization problems introduced by Rössner and Seifert [C. Rössner, J.P. Seifert, The complexity of approximate optima for greatest common divisor computations, in: Proceedings of the 2nd International Algorithmic Number Theory Symposium, ANTS-II, 1996, pp. 307–322]: Given n numbers a 1 , … , a n ∈ Z , find an ℓ ∞ - minimum GCD multiplier for a 1 , … , a n , i.e., a vector x ∈ Z n with minimum max 1 ≤ i ≤ n | x i | satisfying ∑ i = 1 n x i a i = gcd ( a 1 , … , a n ) . We show that assuming P ≠ NP , it is NP-hard to approximate the Minimum GCD Multiplier in ℓ ∞ norm ( GCDM ∞ ) within a factor n c / log log n for some constant c > 0 where n is the dimension of the given vector. This improves on the best previous result. The best result so far gave 2 ( log n ) 1 − ϵ factor hardness by Rössner and Seifert [C. Rössner, J.P. Seifert, The complexity of approximate optima for greatest common divisor computations, in: Proceedings of the 2nd International Algorithmic Number Theory Symposium, ANTS-II, 1996, pp. 307–322], where ϵ > 0 is an arbitrarily small constant.