A factorization question

Number Theory Level 2

Let $N$ be a positive integer ending in 9 such that $N < 50000$ .

Let $a$ and $b$ be positive integers such that $a \times b = N$ and a > b.

Let $c$ and $d$ be positive integers such that $c \times d = 2N$ and $c > d$ .

Is it possible to find a value of $N$ such that it meets the specifications of $\dfrac ab < 1.25$ and $\dfrac cd < 1.02$ ?

No Yes

1 solution

Denton Young
Feb 24, 2016

Let N = 43659

a = 231, b = 189

c= 297, d = 294

How did you get these numbers?

Pi Han Goh - 5 years, 3 months ago

N = 9 x 11 x 21 x 21

a= 9 x 21

b = 11 x 21

2N = 2 x 9 x 11 x 21 x 21

c= 9 x 11 x 3

d= 2 x 3 x 7 x 7

Denton Young - 5 years, 3 months ago

I mean, how did you find these number to begin with?

Pi Han Goh - 5 years, 3 months ago

@Pi Han Goh – Worked them out in my head. You can't use even numbers or a multiple of 5 as any of the factors of N, which restricts the possibilities. In order to have 2N also be factored into two close numbers, your factorization of N has to be one where you can shift by a factor of close to 2. 81/49 came to mind, so that means you need a factor of 3 and a factor of 7 in both a and b. The rest was simple.

Denton Young - 5 years, 3 months ago

A factorization question

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