Problem:
Score : 500 points

Problem Statement

Given are N positive integers A_1,...,A_N.

Consider positive integers B_1, ..., B_N that satisfy the following condition.

Condition: For any i, j such that 1 \leq i < j \leq N, A_i B_i = A_j B_j holds.

Find the minimum possible value of B_1 + ... + B_N for such B_1,...,B_N.

Since the answer can be enormous, print the sum modulo (10^9 +7).

Constraints

1 \leq N \leq 10^4

1 \leq A_i \leq 10^6

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 ... A_N

Output

Print the minimum possible value of B_1 + ... + B_N for B_1,...,B_N that satisfy the condition, modulo (10^9 +7).

Sample Input 1

3
2 3 4

Sample Output 1

13

Let B_1=6, B_2=4, and B_3=3, and the condition will be satisfied.

Sample Input 2

5
12 12 12 12 12

Sample Output 2

5

We can let all B_i be 1.

Sample Input 3

3
1000000 999999 999998

Sample Output 3

996989508

Print the sum modulo (10^9+7).

Solution in Common Lisp (SBCL 1.1.14) (judge verdict: Accepted)

;; -*- coding: utf-8 -*-
(eval-when (:compile-toplevel :load-toplevel :execute)
  (sb-int:defconstant-eqx OPT
    #+swank '(optimize (speed 3) (safety 2))
    #-swank '(optimize (speed 3) (safety 0) (debug 0))
    #'equal)
  #+swank (ql:quickload '(:cl-debug-print :fiveam) :silent t)
  #-swank (set-dispatch-macro-character
           ;; enclose the form with VALUES to avoid being captured by LOOP macro
           #\# #\> (lambda (s c p) (declare (ignore c p)) `(values ,(read s nil nil t)))))
#+swank (cl-syntax:use-syntax cl-debug-print:debug-print-syntax)
#-swank (disable-debugger) ; for CS Academy

;; BEGIN_INSERTED_CONTENTS
;;;
;;; Binomial coefficient with mod
;;; build: O(n)
;;; query: O(1)
;;;

;; TODO: non-global handling

(defconstant +binom-size+ 1000001)
(defconstant +binom-mod+ #.(+ (expt 10 9) 7))

(declaim ((simple-array (unsigned-byte 31) (*)) *fact* *fact-inv* *inv*))
(defparameter *fact* (make-array +binom-size+ :element-type '(unsigned-byte 31))
  "table of factorials")
(defparameter *fact-inv* (make-array +binom-size+ :element-type '(unsigned-byte 31))
  "table of inverses of factorials")
(defparameter *inv* (make-array +binom-size+ :element-type '(unsigned-byte 31))
  "table of inverses of non-negative integers")

(defun initialize-binom ()
  (declare (optimize (speed 3) (safety 0)))
  (setf (aref *fact* 0) 1
        (aref *fact* 1) 1
        (aref *fact-inv* 0) 1
        (aref *fact-inv* 1) 1
        (aref *inv* 1) 1)
  (loop for i from 2 below +binom-size+
        do (setf (aref *fact* i) (mod (* i (aref *fact* (- i 1))) +binom-mod+)
                 (aref *inv* i) (- +binom-mod+
                                   (mod (* (aref *inv* (rem +binom-mod+ i))
                                           (floor +binom-mod+ i))
                                        +binom-mod+))
                 (aref *fact-inv* i) (mod (* (aref *inv* i)
                                             (aref *fact-inv* (- i 1)))
                                          +binom-mod+))))

(initialize-binom)

;;;
;;; Arithmetic operations with static modulus
;;;

;; FIXME: Currently MOD* and MOD+ doesn't apply MOD when the number of
;; parameters is one.
(defmacro define-mod-operations (divisor)
  `(progn
     (defun mod* (&rest args)
       (reduce (lambda (x y) (mod (* x y) ,divisor)) args))

     (sb-c:define-source-transform mod* (&rest args)
       (if (null args)
           1
           (reduce (lambda (x y) `(mod (* ,x ,y) ,',divisor)) args)))

     (defun mod+ (&rest args)
       (reduce (lambda (x y) (mod (+ x y) ,divisor)) args))

     (sb-c:define-source-transform mod+ (&rest args)
       (if (null args)
           0
           (reduce (lambda (x y) `(mod (+ ,x ,y) ,',divisor)) args)))

     (define-modify-macro incfmod (delta)
       (lambda (x y) (mod (+ x y) ,divisor)))

     (define-modify-macro decfmod (delta)
       (lambda (x y) (mod (- x y) ,divisor)))

     (define-modify-macro mulfmod (multiplier)
       (lambda (x y) (mod (* x y) ,divisor)))))

(macrolet ((def (name fname)
             `(define-modify-macro ,name (new-value) ,fname)))
  (def minf min)
  (def maxf max)
  (def mulf *)
  (def divf /)
  (def iorf logior)
  (def xorf logxor)
  (def andf logand))

(declaim (ftype (function * (values fixnum &optional)) read-fixnum))
(defun read-fixnum (&optional (in *standard-input*))
  "NOTE: cannot read -2^62"
  (declare #.OPT)
  (macrolet ((%read-byte ()
               `(the (unsigned-byte 8)
                     #+swank (char-code (read-char in nil #\Nul))
                     #-swank (sb-impl::ansi-stream-read-byte in nil #.(char-code #\Nul) nil))))
    (let* ((minus nil)
           (result (loop (let ((byte (%read-byte)))
                           (cond ((<= 48 byte 57)
                                  (return (- byte 48)))
                                 ((zerop byte) ; #\Nul
                                  (error "Read EOF or #\Nul."))
                                 ((= byte #.(char-code #\-))
                                  (setf minus t)))))))
      (declare ((integer 0 #.most-positive-fixnum) result))
      (loop
        (let* ((byte (%read-byte)))
          (if (<= 48 byte 57)
              (setq result (+ (- byte 48)
                              (* 10 (the (integer 0 #.(floor most-positive-fixnum 10))
                                         result))))
              (return (if minus (- result) result))))))))

(declaim (ftype (function * (values simple-bit-vector &optional)) make-prime-table))
(defun make-prime-table (sup)
  "Returns a simple-bit-vector of length SUP, whose (0-based) i-th bit is 1 if i
is prime and 0 otherwise.

Example: (make-prime-table 10) => #*0011010100"
  (declare (optimize (speed 3) (safety 0)))
  (check-type sup (integer 2 (#.array-total-size-limit)))
  (let ((table (make-array sup :element-type 'bit :initial-element 0))
        (sup/64 (ceiling sup 64)))
    ;; special treatment for p = 2
    (dotimes (i sup/64)
      (setf (sb-kernel:%vector-raw-bits table i) #xAAAAAAAAAAAAAAAA))
    (setf (sbit table 1) 0
          (sbit table 2) 1)
    ;; p >= 3
    (loop for p from 3 to (+ 1 (isqrt (- sup 1))) by 2
          when (= 1 (sbit table p))
          do (loop for composite from (* p p) below sup by p
                   do (setf (sbit table composite) 0)))
    table))

;; FIXME: Currently the element type of the resultant vector is (UNSIGNED-BYTE 62).
(declaim (ftype (function * (values (simple-array (integer 0 #.most-positive-fixnum) (*))
                                    simple-bit-vector
                                    &optional))
                make-prime-sequence))
(defun make-prime-sequence (sup)
  "Returns the ascending sequence of primes smaller than SUP."
  (declare (optimize (speed 3) (safety 0)))
  (check-type sup (integer 2 (#.array-total-size-limit)))
  (let ((table (make-prime-table sup)))
    (let* ((length (count 1 table))
           (result (make-array length :element-type '(integer 0 #.most-positive-fixnum)))
           (index 0))
      (declare ((integer 0 #.most-positive-fixnum) length))
      (loop for x below sup
            when (= 1 (sbit table x))
            do (setf (aref result index) x)
               (incf index))
      (values result table))))

(defstruct (prime-data (:constructor %make-prime-data (seq table)))
  (seq nil :type (simple-array (integer 0 #.most-positive-fixnum) (*)))
  (table nil :type simple-bit-vector))

(defun make-prime-data (sup)
  (multiple-value-call #'%make-prime-data (make-prime-sequence sup)))

(declaim (inline factorize)
         (ftype (function * (values list &optional)) factorize))
(defun factorize (x prime-data)
  "Returns the associative list of prime factors of X, which is composed
of (<prime> . <exponent>). E.g. (factorize 100 <prime-table>) => '((2 . 2) (5
. 5)).

- Any numbers beyond the range of PRIME-DATA are regarded as prime.
- The returned list is in descending order w.r.t. prime factors."
  (declare (integer x))
  (setq x (abs x))
  (when (<= x 1)
    (return-from factorize nil))
  (let ((prime-seq (prime-data-seq prime-data))
        result)
    (loop for prime of-type unsigned-byte across prime-seq
          do (when (= x 1)
               (return-from factorize result))
             (loop for exponent of-type (integer 0 #.most-positive-fixnum) from 0
                   do (multiple-value-bind (quot rem) (floor x prime)
                        (if (zerop rem)
                            (setf x quot)
                            (progn
                              (when (> exponent 0)
                                (push (cons prime exponent) result))
                              (loop-finish))))))
    (if (= x 1)
        result
        (cons (cons x 1) result))))

(defmacro dbg (&rest forms)
  #+swank
  (if (= (length forms) 1)
      `(format *error-output* "~A => ~A~%" ',(car forms) ,(car forms))
      `(format *error-output* "~A => ~A~%" ',forms `(,,@forms)))
  #-swank (declare (ignore forms)))

(defmacro define-int-types (&rest bits)
  `(progn
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "UINT~A" b)) () '(unsigned-byte ,b))) bits)
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "INT~A" b)) () '(signed-byte ,b))) bits)))
(define-int-types 2 4 7 8 15 16 31 32 62 63 64)

(declaim (inline println))
(defun println (obj &optional (stream *standard-output*))
  (let ((*read-default-float-format* 'double-float))
    (prog1 (princ obj stream) (terpri stream))))

(defconstant +mod+ 1000000007)

;;;
;;; Body
;;;

(define-mod-operations +mod+)
(defun main ()
  (declare #.OPT)
  (let* ((n (read))
         (pdata (make-prime-data 10000))
         (as (make-array n :element-type 'uint31))
         (lcm-table (make-hash-table :size 1000 :test #'eq)))
    (dotimes (i n)
      (setf (aref as i) (read-fixnum))
      (dolist (node (factorize (aref as i) pdata))
        (let ((prime (car node))
              (exp (cdr node)))
          (declare (uint31 prime exp))
          (if (gethash prime lcm-table)
              (maxf (the uint31 (gethash prime lcm-table)) exp)
              (setf (gethash prime lcm-table) exp)))))
    (let ((lcm 1))
      (declare (uint31 lcm))
      (maphash (lambda (prime exp)
                 (declare (uint31 prime exp))
                 (dotimes (_ exp)
                   (mulfmod lcm prime)))
               lcm-table)
      (let ((res 0))
        (declare (uint62 res))
        (loop for a across as
              do (incf res (mod* lcm (aref *inv* a))))
        (println (mod res +mod+))))))

#-swank (main)

;;;
;;; Test and benchmark
;;;

#+swank
(defun io-equal (in-string out-string &key (function #'main) (test #'equal))
  "Passes IN-STRING to *STANDARD-INPUT*, executes FUNCTION, and returns true if
the string output to *STANDARD-OUTPUT* is equal to OUT-STRING."
  (labels ((ensure-last-lf (s)
             (if (eql (uiop:last-char s) #\Linefeed)
                 s
                 (uiop:strcat s uiop:+lf+))))
    (funcall test
             (ensure-last-lf out-string)
             (with-output-to-string (out)
               (let ((*standard-output* out))
                 (with-input-from-string (*standard-input* (ensure-last-lf in-string))
                   (funcall function)))))))

#+swank
(defun get-clipbrd ()
  (with-output-to-string (out)
    ;; (run-program "C:/Windows/System32/WindowsPowerShell/v1.0/powershell.exe" '("get-clipboard") :output out)
    (run-program "powershell.exe" '("-Command" "Get-Clipboard") :output out :search t)))

#+swank (defparameter *this-pathname* (uiop:current-lisp-file-pathname))
#+swank (defparameter *dat-pathname* (uiop:merge-pathnames* "test.dat" *this-pathname*))

#+swank
(defun run (&optional thing (out *standard-output*))
  "THING := null | string | symbol | pathname

null: run #'MAIN using the text on clipboard as input.
string: run #'MAIN using the string as input.
symbol: alias of FIVEAM:RUN!.
pathname: run #'MAIN using the text file as input."
  (let ((*standard-output* out))
    (etypecase thing
      (null
       (with-input-from-string (*standard-input* (delete #\Return (get-clipbrd)))
         (main)))
      (string
       (with-input-from-string (*standard-input* (delete #\Return thing))
         (main)))
      (symbol (5am:run! thing))
      (pathname
       (with-open-file (*standard-input* thing)
         (main))))))

#+swank
(defun gen-dat ()
  (uiop:with-output-file (out *dat-pathname* :if-exists :supersede)
    (format out "")))

#+swank
(defun bench (&optional (out (make-broadcast-stream)))
  (time (run *dat-pathname* out)))

;; To run: (5am:run! :sample)
#+swank
(it.bese.fiveam:test :sample
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "3
2 3 4
"
    "13
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "5
12 12 12 12 12
"
    "5
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "3
1000000 999999 999998
"
    "996989508
")))