Solution

    PROGRAM Test_Outer
     IMPLICIT NONE
      REAL, ALLOCATABLE, DIMENSION(:) :: A
      REAL, ALLOCATABLE, DIMENSION(:) :: B
      INTEGER N_elts, istat
    
       PRINT*, "Type in size of A vector"
       READ*, N_elts
       ALLOCATE(A(N_elts), STAT=istat)
       IF (istat .NE. 0) THEN
        PRINT*, "Allocation Error"
        STOP
       END IF
    
       PRINT*, "Type in size of B vector"
       READ*, N_elts
       ALLOCATE(B(N_elts), STAT=istat)
       IF (istat .NE. 0) THEN
        PRINT*, "Allocation Error"
        STOP
       END IF
    
       CALL RANDOM_NUMBER(A)
       CALL RANDOM_NUMBER(B)
     
       PRINT*, "Outer product of A and B is:"
       PRINT*, Outer(A,B)
     
       DEALLOCATE(A)
       DEALLOCATE(B)

    CONTAINS
    
     FUNCTION Outer(A,B)
      REAL, INTENT(IN), DIMENSION(:) :: A, B
      REAL, DIMENSION(SIZE(A),SIZE(B)) :: Outer
      INTEGER i,j
    
       DO i = 1, SIZE(A)
        DO j = 1, SIZE(B)
         Outer(i,j) = A(i) * B(j)
        END DO
       END DO

     END FUNCTION Outer
    
    END PROGRAM Test_Outer