Time For Proof By Exhaustion

Number Theory Level 1

Which of the following statements are true?

(A) : The sum of any 2 consecutive integers is divisible by 2.
(B) : The sum of any 3 consecutive integers is divisible by 3.
(C) : The sum of any 4 consecutive integers is divisible by 4.
(D) : The sum of any 5 consecutive integers is divisible by 5.

(C) and (D) only (B) and (D) only (A) and (C) only (A) and (B) only

Pi Han Goh by Pi Han Goh , 30, Malaysia

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1 solution

Let m m be an integer.

  • The first statement can be expressed as m + ( m + 1 ) = 2 m + 1 m+(m+1)=2m+1 , which is odd, thus, is not divisible by 2.

  • The second is the same that m + ( m + 1 ) + ( m + 2 ) = 3 m + 3 = 3 ( m + 1 ) m+(m+1)+(m+2)=3m+3=3(m+1) , which clearly is divisible by 3.

  • m + ( m + 1 ) + ( m + 2 ) + ( m + 3 ) = 4 m + 6 m+(m+1)+(m+2)+(m+3)=4m+6 , which is not divisible by 4, because 4 ∣ 4 m 4|4m , but 4 ∤ 6 4\nmid 6 .

  • m + ( m + 1 ) + ( m + 2 ) + ( m + 3 ) + ( m + 4 ) = 5 m + 10 = 5 ( m + 2 ) m+(m+1)+(m+2)+(m+3)+(m+4)=5m+10=5(m+2) , which is divisible by 5.

Therefore, only (B) and (D) are true.

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Im still confused

Sharif Tarver - 3 years, 1 month ago

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