Arithmetic: answer key to the exercises in the final year of high school in PDF.

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Exercise of arithmetic in terminale S.

1-Position d = gcd(a,b)

We have if d divides a and d divides b then d divides b and d divides (a-bq)

Reciprocally: if d divides b and d divides (a-bq) then d divides ( a – bq ) +bq = a

2- it is the previous relationship with $a\,=\,5n^3\,-n\,;\,b\,=\,n+2\,;\,q\,=\,5n^2-\,10n\,+19$

Demonstrate a property, exercise of mathematics in terminale S on arithmetic.

Let δ = PGCD(a ;b) and µ = PPCM(a ;b).

Then we have a= δa’ and b = δb’ with a’ and b’ prime between them.

So we have PPCM(a’ ;b’) = a’b’

µ = PPCM(δa’; δb’) = δ×PPCM(a’; b’) = δ×a’×b’

Thus δµ = δ²×a’×b’ = δ×a’× δ×b’ = ab

System of equations and arithmetic.

From the relation PPCM(a;b)×PGCD(a;b) = ab,

we deduce from the second equation of the system :

3×PPCM(a ;b) = PGCD(a ;b)×PPCM(a ;b)

So PGCD(a;b) = 3

Then there are integers a’ and b’ (prime to each other) such that a = 3a’ and b = 3b’.

Carrying over into the first equation, we get :

9a’² – 9b’² = 405

Let (a’ + b’)(a’ – b’) = 45

Now 45 = 3²×5

We can therefore have the following 3 systems:

$\,\{\,a'\,+\,b'\,=\,45\\\,a'\,-b'\,=\,1\,.$ $\,\{\,a'\,+\,b'\,=\,15\\\,a'\,-b'\,=\,3\,.$ $\,\{\,a'\,+\,b'\,=\,9\\\,a'\,-b'\,=\,5\,.$

a’ = 23 and b’ = 22 a’ = 9 and b’ = 6 a’ = 7 and b’ = 2

Hence (a ;b) = (66 ;69) is not suitable because hence (a ;b) = (42 ;6)

9 and 6 are not prime to each other

The solution pairs of the system are therefore (42 ;6) and (66 ;69).

Prime numbers .

1) $(a^2\,+\,2b^2)^2\,=\,a^4\,+\,4a^2b^2\,+\,4b^4$

2) $a^4+\,4b^4\,=\,(a^2\,+\,2b^2)^2\,-\,4a^2b^2\,=\,(a^2\,+\,2b^2\,+\,2ab)(a^2\,+\,2b^2\,-\,2ab)$

$a^4+\,4b^4=\,%5B(a\,+\,b)^2\,+\,b^2%5D%5B(a\,-\,b)^2\,+\,b^2%5D$

$(a\,+\,b)^2\,+\,b^2\geq\,\,4\,et\,(a\,-\,b)^2\,+\,b^2\geq\,\,4.$

because a and b are greater than or equal to 2.

is the product of two integers greater than 1.

is not a prime number.

Exercise:

The exercises in the final year

After having consulted the answer key of this exercise arithmetic in senior year, you can return to the exercises in senior year

Senior year exercises.

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