# euler.lisp @ 2e707232cee0

`Problem 5`
author Steve Losh Sun, 10 Apr 2016 01:27:54 +0000 5dc80bffbecc 37bd082d9946
```(in-package #:euler)

;;;;
(defun digits (n)
"Return how many digits `n` has in base 10."
(values (truncate (1+ (log n 10)))))

(defun definitely-palindrome-p (n)
"Return whether `n` is a palindrome (in base 10), the slow-but-sure way."
(let ((s (format nil "~D" n)))
(string= s (reverse s))))

(defun palindrome-p (n)
"Return whether `n` is a palindrome (in base 10)."
(assert (>= n 0) (n) "~A must be a non-negative integer" n)
;; All even-length base-10 palindromes are divisible by 11, so we can shortcut
;; the awful string comparison. E.g.:
;;
;;   abccba =
;;   100001 * a +
;;   010010 * b +
;;   001100 * c
(cond
((zerop n) t)
((and (evenp (digits n))
(not (dividesp n 11))) nil)
(t (definitely-palindrome-p n))))

(defun range (from below)
(loop :for i :from from :below below
:collect i))

;;;; Problems
(defun problem-1 ()
;; If we list all the natural numbers below 10 that are multiples of 3 or 5,
;; we get 3, 5, 6 and 9. The sum of these multiples is 23.
;;
;; Find the sum of all the multiples of 3 or 5 below 1000.
(loop :for i :from 1 :below 1000
:when (or (dividesp i 3)
(dividesp i 5))
:sum i))

(defun problem-2 ()
;; Each new term in the Fibonacci sequence is generated by adding the previous
;; two terms. By starting with 1 and 2, the first 10 terms will be:
;;
;;     1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
;;
;; By considering the terms in the Fibonacci sequence whose values do not
;; exceed four million, find the sum of the even-valued terms.
(loop :with p = 0
:with n = 1
:while (<= n 4000000)
:when (evenp n) :sum n
:do (psetf p n
n (+ p n))))

(defun problem-3 ()
;; The prime factors of 13195 are 5, 7, 13 and 29.
;;
;; What is the largest prime factor of the number 600851475143 ?
(apply #'max (prime-factorization 600851475143)))

(defun problem-4 ()
;; A palindromic number reads the same both ways. The largest palindrome made
;; from the product of two 2-digit numbers is 9009 = 91 × 99.
;;
;; Find the largest palindrome made from the product of two 3-digit numbers.
(let ((result (list)))
(loop :for i :from 0 :to 999
:do (loop :for j :from 0 :to 999
:for product = (* i j)
:when (palindrome-p product)
:do (push product result)))
(apply #'max result)))

(defun problem-5 ()
;; 2520 is the smallest number that can be divided by each of the numbers from
;; 1 to 10 without any remainder.
;;
;; What is the smallest positive number that is evenly divisible by all of the
;; numbers from 1 to 20?
(let ((divisors (range 11 21)))
;; all numbers are divisible by 1 and we can skip checking everything <= 10
;; because:
;;
;; anything divisible by 12 is automatically divisible by 2
;; anything divisible by 12 is automatically divisible by 3
;; anything divisible by 12 is automatically divisible by 4
;; anything divisible by 15 is automatically divisible by 5
;; anything divisible by 12 is automatically divisible by 6
;; anything divisible by 14 is automatically divisible by 7
;; anything divisible by 16 is automatically divisible by 8
;; anything divisible by 18 is automatically divisible by 9
;; anything divisible by 20 is automatically divisible by 10
(loop :for i
:from 20 :by 20 ; it must be divisible by 20
:when (every (lambda (n) (dividesp i n))
divisors)
:return i)))```