Mathematical Proofs

Edexcel IAL WMA12/01/P2/Oct 2022/Q1 (Proof: Exhaustion)

Given that a, b and c are integers greater than 0 such that

• c = b + 2

• a + b + c = 10

Prove, by exhaustion, that the product of a, b and c is always even.

You may use the table below to illustrate your answer.​​ 

(3)​​ 

You may not need to use all rows of this table.

SOLUTION​​ 

When​​ b=1​​ (given in question)

c=b+2=1+2

c=3

a+b+c=10

a+1+3=10

a=6

a×b×c=6×1×3=18 (even)

When​​ b=2​​ (given in question)

c=b+2=2+2

c=4

a+b+c=10

a+2+4=10

a=4

a×b×c=4×2×4=32 (even)

When​​ b=3​​ (continuing the pattern in the question)

c=b+2=3+2

c=5

a+b+c=10

a+3+5=10

a=2

a×b×c=2×3×5=30 (even)

When​​ b=4​​ (continuing the pattern in the question)

c=b+2=4+2

c=6

a+b+c=10

a+4+6=10

a=0

a×b×c=0×4×6=0 (even)

Hence, it is proved by exhaustion that for all possible value of a, b, and c, the product of a, b and c is always even.