We present a new simple and efficient quantum algorithm for factoring composite integers. The key step consists of making use of the new exponentially fast search algorithm [1]. Let $M = PQ$, $0 < M < 2^n$, be a composite integer with two nearly equal prime factors $P,Q$ to be determined. We prepare register, say $A = A_1 \otimes A_2$, where $A_1$ and $A_2$ are sub-registers containing equally weighted superposition of states $|x\rangle$ and $|y\rangle$ respectively such that $0 \le x,y \le 2^n - 1$. Register $A$ thus contains equally weighted superposition of the tensor product states $|x\rangle|y\rangle$. We also prepare register $B$ to keep the images of elements in register $A$ produced under the action of the function, $f$, $f : A \to B$, defined as $f(|x\rangle|y\rangle) = |xy\rangle$. The transformation, $U_f$, corresponding to the given function $f$ is defined as $U_f : (|x\rangle|y\rangle)_A(|0\rangle)_B \to (|x\rangle|y\rangle)_A(|xy\rangle)_B$, which can also take superposition of states as input. Thus, register $A$ is entangled with register $B$ through $U_f$. After building up quantum registers $A$ and $B$ we operate on register $B$ a unitary operator, $U$, defined as $U = \sum_z |v\rangle\langle v|$, where $0 \le z \le 2^n - 1$, and $|v\rangle = |00\cdots0\rangle_n|z\rangle$. The action of $U$ will change the state in register $B$ into the uniform superposition state $\frac{1}{\sqrt{2^n}} \sum_{z=0}^{2^n-1} |v\rangle$, so that this state now becomes separable and the algorithm in [1] becomes applicable for searching a target state present as a component state in this modified register $B$. We then perform the key step of applying [1] on register $B$ for searching the target state $|t\rangle = |00\cdots0\rangle_n|M\rangle$, and amplify to the full extent the amplitude of this target state so that if we now partially measure the register $B$ it will almost always be found in this target state $|t\rangle = |00\cdots0\rangle_n|M\rangle$. As a consequence of this the state in register $A$ now changes into the superposition state $\frac{1}{\sqrt{2}} (|1\rangle|M\rangle + |P\rangle|Q\rangle + |Q\rangle|P\rangle + |M\rangle|1\rangle)$. This new state in register $A$ now clearly contains the desired prime factorization which is easily revealed after further partial measurement done on any one of the sub-registers $A_1,A_2$.