2048!!!

Number Theory Level 1

Represent 2048 as the product of two positive numbers such that one number is twice the other. Find them and enter as the sum of the two numbers.

The answer is 96.

4 solutions

Mahdi Raza
Jun 21, 2020

One number is $a$ , the other is twice of that which is $2a$ . Given that $2048 = a \cdot 2a \implies a = 32$ . Thus $a + 2a = \boxed{96}$

I used a different approach, which is probably messier than yours, but it works. 2048 = 2^11, so the numbers must be 2^5 and 2^6 as 2^6 is 2 times of 2^5. 2^6 = 64, 2^5 = 32, 64+32 = 96. I clicked Brilliant on your solution too.

A Former Brilliant Member - 11 months, 3 weeks ago

Thanks. Your solution is nice as well!

Mahdi Raza - 11 months, 3 weeks ago

Thanks, check out the problem i posted in rocket fuel reaction.

A Former Brilliant Member - 11 months, 3 weeks ago

Alvin Willio
Sep 27, 2014

2048 is the product of two positive numbers is equivalent with

$2048=a.b$

and one number is twice the other is equivalent with

$a=2b$

then we get

$2048=2{ b }.b$

$2048=2{ b }^{ 2 }$

$1024={ b }^{ 2 }$

$b=32$

substitute $b=32$ into $a=2b$

so, we get $a=64$

finally, we know that $a+b=96$

Gabriel Serpa
Aug 22, 2014

a>0 , b>0, b=2a

a.b=2048

2a^2=2^11

a^2=2^10

a=2^5 => a=32 , b=64

a+b=32+64

a+b=96

Ubair Khan
Aug 11, 2014

The number 2048 should directly give you the idea that the two numbers asked must be the results of '2^n', for two different values of n obviously. So, 2048 = 32*64 32+64 = 96

2048!!!

The answer is 96.

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