Which of the following statements is true?
I. The product of 3 consecutive numbers is divisible by 23 .
II. The product of 11 consecutive numbers is divisible by 22 .
III. The product of 7 consecutive numbers is divisible by 29 .
IV. The product of 5 consecutive numbers is divisible by 37 .
V. The product of 10 consecutive numbers is divisible by 11 .
A) IV
B) V
C) III
D) II
E) I

Question

Which of the following statements is true? ​ 
I. The product of 3 consecutive numbers is divisible by 23 . 
II. The product of 11 consecutive numbers is divisible by 22 . 
III. The product of 7 consecutive numbers is divisible by 29 . 
IV. The product of 5 consecutive numbers is divisible by 37 . 
V. The product of 10 consecutive numbers is divisible by 11 . ​
A) IV
B) V
C) III
D) II
E) I

Quizplus · Accepted Answer

I. Not true as there exist three consecutive numbers such that their product is not divisible by 23, for example, 3 x 4 x 5 = 60 which is not divisible by 23.
II. Not true as the product of 11 consecutive numbers contains a factor of 11 and a factor of 2, but not necessarily a factor of 22.
III. Not true as there exist seven consecutive numbers such that their product is not divisible by 29, for example, 1 x 2 x 3 x 4 x 5 x 6 x 7 = 5040 which is not divisible by 29.
IV. True as it can be proven that for any five consecutive numbers, at least one of them must be divisible by 37 (use pigeonhole principle), hence their product is divisible by 37.
V. True as 10 consecutive numbers contains a factor of 11, hence their product is divisible by 11.

Which of the Following Statements Is True? ​

Which of the Following Statements Is True?