Mathematical Proofs

WMA12/01 IAL (Edexcel) P2/June 2021/Q3 (Proof by Exhaustion, Deduction)

• Prove that for all single digit prime numbers, p,​​ 

p3+p​​ is a multiple of 10

(2)

• Show, using algebra, that for​​ n  N​​ 

n + 13 – n3 is not a multiple of 3 

(3)

SOLUTION

i-

We need to prove that the statement given to us is true for all digit prime numbers, p. We will substitiute the prime numbers in place of p one by one in ascending order. ​​​​ 

P=2;     P3+P=23+2=8+2=10 , 10=10 x 1

P=3;   P3+P=33+3=27+3=30 , 30=10 x 3

P=5;  P3+P=53+5=127+5=130 , 130=10 x 13

P=7; P3+P=73+7=343+7=350 , 350=10 x 35

All single digit prime number given a multiple of ten when substituted into​​ P3+p​​ by exhaustion the statement is true.

ii-​​ Using algeabraic method​​ 

n+13-n3=n3+3n2+3n+1-n3

=3n2+3n+1

=3n2+n+1

Though​​ 3n2+n​​ is a multiple of 3 but​​ 3n2+n+1​​ is 1 more than a multiple of 4.​​ 

Hence,​​ 3n2+n+1 is not a multiple of​​ 3.​​