Mathematical Proofs

Edexcel IAL WMA12/01/P2/Jan 2022/Q10 (Proof: Counter Example, Deduction & Exhaustion)

• Prove by counter example that the statement​​ 

“if p is a prime number then 2p + 1 is also a prime number”​​ 

is not true.​​ 

(1)

• Use proof by exhaustion to prove that if n is an integer then​​ 

5n2 + n + 12 

is always even.​​ 

(4)

SOLUTION

a- We need one such example that can negate the statement. We will substitiute the prime numbers in place of p one by one in ascendinf order. ​​ 

p=2    ⇒  2p+1=22+1=5 (prime number)

p= 3   ⇒  2p+1=23+1=7 (prime number)

p=5    ⇒  2p+1=25+1=11 (prime number)

p=7    ⇒  2p+1=27+1=15 (not a prime number)

Since 15 has more than 2 factors (1, 3, 5, & 15), it is not a prime number.​​ 

So, the statement is not correct.

b-​​ 

We will split all the integers in odd​​ n=2k+1 and even numbers (n=2k), where​​ k∈integer.

Let us first consider all even numbers:​​ n=2k.

5n2+n+12 =5 2k2+2k+12

=54k2+2k+12

=20k2+ 2k+12

=210k2+k+6

Since this expression is a multiple of 2 so it is an even number.​​ 

=2 x integer (even)

∴n is even, 5n2+n+16 is also even.

Now, Lets consider all odd numbers:​​ n=2k+1 

5n2+n+12=52k+12+2k+1+12

=54k2+4k+1+2k+13

=20k2+20k+5+2k+13

=20k2+22k+18

=210k2+11k+9

This expression is also a multiple of 2, so it is an even number.

Hence, it can be concluded that for all integers whether even or odd,​​ 5n2 + n + 12 will always be even.​​