Number questions

Question

If a and b are positive integers and 24ab is a perfect square, then which of the following cannot be possible? 

1. Both a and b are odd

2. ab is a perfect square 

3. Both a and b are divisible by 6 

Multiple answres are possible 

Answer 1

"24ab is a perfect square"

   Let x=ab

 

24x=2*2*2*3*x is a perfect square, so,

        x = 2*3*y   where y is also a positive integer.

     That makes 24x = 2*2*2*3*2*3*y a perfect square.

 

Consider:

1. Both a and b are odd

    We know x is even (x = 2*3*y), but if both a and b are odd, ab is also odd.  [1] cannot be true.

2. ab is a perfect square 

     From above, x=ab=2*3*y.  For this to be a perfect square, it requires y=2*3*z, so ab=2*3*2*3*z and z must be a perfect square!.  [2] might or might not be true [it is only true if 36 is a factor of ab and z is a perfect square, for example if x=ab=36*4=144, so ab is a perfect square.

 

3. Both a and b are divisible by 6 

     That means a=6m and b=6n (where m and n are positive integers).

     x = ab = 36mn  is possible if the product mn has an even number of the prime factors (so as to make x=ab a perfect square).  [3] is possible, for example x=ab=36*4=144 (where m=2 and n=2, and a=12 and b=12, so a and b are divisible by 6).

 

Only [1] cannot be possible.

Answer 2

24ab=2*2*2*3*a*b

 

to be a perfect square each prime number must occur an even number of times

we need a 2 and a 3 to make a perfect square (the smallest one, given the number 24)

the 2 can be a factor of another number

a and b both cannot be odd, which is 1.

 

ab cannot be a perfect square because 24ab is 24*ab and a perfect square times another perfect square gives you a third perfect square, so if 24ab is a perfect square, then ab cannot be a perfect square which is 2.

 

3. is possible

24ab=2*2*2*3*a*b where a=2 and b=3 gives a perfect square

multiply a=2 by 9 to get 18 and multiply 3 by 4 to get 12

12 and 18 are both divisible by 6

there are many other possibilities

a has to be at least 2 and b has to be at least 3 to get a perfect square

now multiply the 2 by another perfect square (such as 9) so that the result is divisible by 6

also multiply the 3 by another perfect square (such as 4) so that the result is divisible by 6

 

Answer 3

24 = 2³3

 

24ab can be written as a product of primes of only non-negative even powers since it's a perfect square (take a prime decomposition of its square root then combine that with itself).

 

We must be able to write ab as a positive odd power of 2 because of the 2³ in how we wrote 24. An odd number is an even plus an odd always, so one of a and b has 2 to an even power, and one has it to an odd power. The one having it to an odd power is even, since 1 is the smallest positive odd number

 

Likewise, we must be able to write ab as a positive odd power of 3. Similar reasoning applies.

 

I will leave the details for you, but the answer should be options 1 and 2 are not possible but option 3 is possible (give an example of values of a and b that satisfy 3)