Cours et Exercices

Back to the exercise 1)  a= x²-5x+4= x²- 2 x +1 - 3 x +3 =(x - 1 ) ² - 3 (x- 1) =(x - 1 )(x - 1 - 3 )= (x - 1 )(x - 4 ) b =2 x²+3x-2 = 2 x²+4x- x -2 =2x  (x +2 )-(x +2 ) = (x +2 )(2 x - 1 )   2)  3)  4) the product of two consecutive integers is a natural even number Back to the exercise

Correction 1) factorize the following expression : a= x²-5x+4  b= 2x²+3x-1 2) decompose 3240 and 1440 into a product of primes numbers 3) conclude a simple expression for the two numbers ( the square root of 3240 and 1440 ) 4) k is a natural number , show that k ( k +1) is an even number Correction

Back to the exercise 1) we have for all primes numbers greater than 3 is are odd number , so there is an integer k for p = 2k +1 then : 2) we have and p is a prime number  greater than 3 ,so there are tow integers numbers a and b deux nombres entiers naturels such as and  : While a and b are consecutive   Back to the exercise

Correction 1) p is a prime number greater than 3 Set out : :          a) verify that a and b are integers natural          b) calculate: as a function of p 2) show that every prime number p greater than 3 , is written as a subtraction of two consecutive integers natural Give this subtraction for P =41 Correction

Back to the exercise 1) 49 is not a prime number because may be divided by 7 we have the following numbers   2  ،  3  ،  5  ،  7  ،  11   ،  13  et 17 do not divide 239 and we have 17²>239>23² then 239 is a prime number ....................................................... 2) the decomposition into prime numbers :     Back to the exercise

correction 1) identify the primes numbers of the following numbers : 49  ،  239   ،   407  ،   387  ،   1559  ،    8367  2) decompose the following numbers into prime numbers :   675  ،  1650  ،      5292    ،    6250     correction

Back to the exercise 1) showing that a is a multiple of 3: we have: Back to the exercise

correction Let a and b tow naturels integers numbers Write out the following equations 1) shows that a is a multiple of 3 . 2) show that b can be divided on 12 3) gives the decomposition into prime numbers of a and b 4) calculate the PGCD(a,b)    correction