2018 NSW: Maths Puzzle 5

5th puzzle by Dr Harry Wiggins

A teacher wrote a VERY LARGE number on the board and then he wrote 60 lines.

The 1st line is: "The number is divisible by 2.
The 2nd line is: "The number is divisible by 3. 
The 3rd line is: "The number is divisible by 4.
(and so on)
The 60th line is: "The number is divisible by 61.    
The teacher then commented that exactly two consecutive lines is incorrect.
Which two lines is incorrect or false?
Leave your comment
8/2/2018 3:53 PM
Line 30 and 31

This is because 31 is a prime number which is greater than 30 and 32 is 2^5.

2^5 doesn't divide any other number from 2 to 61, so that line isn't necessarily true, and there are no other multiples of 31 from 2 to 61, do that line isn't necessarily true either.

Since those 2 lines are the only consecutive lines that satisfy the condition, we know that line 30 and 31 are the 2 lines.
8/2/2018 5:27 PM
Let each line L be associated with its supposed factor P.
We know that L=P-1, for any n.
If a line L is false, then its associated P is also called false.
If two consecutive L's are false, and are the only L's that are false, their respective P's cannot have multiples that are other P's. Since all P's from 2 to 30, have multiples less than 61. Thus the two false P's are greater than 30.
If a P is a product of two co-prime numbers greater than 1, it cannot be false as those co-prime factors would also be false, since if a|n, b|n, and a and b are co-prime, ab|n. Thus a false P cannot be a product of two co-prime numbers greater than 1, and is thus a power of a prime number. Since one false P is even, and is a number from 30 to 61, it must be 32, as that is the only power of 2 in that range. The other false P must be either 31 or 33, as they are consecutive. But it cannot be 33, as that is a product of 3 and 11 which are co-prime. Thus the other false P is 31 which is prime. Since the two false P's are 31 and 32, the two false lines are 30 and 31.
Sachin Reddy Northwood School
8/4/2018 5:30 AM
Well done Andi and Sachin for identifying that the number is not divisible by 31 and 32.

Followup: Can you give an example of a number with this property? (Hint: Use factorial!)