## Can anyone help me finding the solution

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aftab
Posts: 5
Joined: Fri Dec 23, 2011 6:45 pm

### Can anyone help me finding the solution

I have been trying to solve a problem for many days. That is-

The product of a man’s son’s age is 1664. The age of the youngest son is more than half the eldest son’s age. The man’s age is 50. Then how many sons does he have?

Problem Analysis:

1. We will have to express 1664 as a product of some small numbers.
2. Every number must be the factor of 1664.
3. Every number must be less than 50.
4. The smallest number must be more than the half of the biggest number.

Finding the solution:

Let us find the prime factors of 1664.

1664 = 2x2x2x2x2x2x2x13
1664 = 4x32x13 This does not fulfill condition 4.
1664 = 8x16x13 This does not fulfill condition 4.
1664 = 4x8x4x13 This does not fulfill condition 4.
1664 = 4x16x26 This does not fulfill condition 4.

Can anyone help me finding the solution?

Labib
Posts: 411
Joined: Thu Dec 09, 2010 10:58 pm
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### Re: Can anyone help me finding the solution

First, notice that,
$1664=2^7.13$

If the man has $i$ sons with their ages $x_1, x_2, ....., x_i,[x_1\leq x_2\leq..... \leq x_i]$.
then, $1\leq x_k \leq 50$, where $1\leq k\leq i$.

If, $x_1=2^m,x_i=2^n,[m<n]$,
$\frac {x_1}{x_i}=\frac 1 {2^{n-m}}\leq \frac 1 2$.

So, one of $x_1$ and $x_i$ would be a multiple of $13$.

Again, as $2^2.13=52$,
one of the two sons may be aged $2.13=26$ years.
So, Now we have two cases.
(a) $x_1=26$.
(b)$x_i=26$.

If, (b) is true, $13<x_1<26 \Rightarrow x_1=16$.
So, there exists $x_k$ such that $x_k<x_1$, a contradiction!!
So, (b) is not true.

If, (a) is true,
$x_1=26\Rightarrow x_i=32$.
This implies that the man has $2$ sons.

In the same way, deal with, $x_1=13$ and $x_i=13$.
You'll find contradictions and will be able to deduce that,

the man has two sons.
Please Install $L^AT_EX$ fonts in your PC for better looking equations,
Learn how to write equations, and don't forget to read

"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." - Sherlock Holmes

aftab
Posts: 5
Joined: Fri Dec 23, 2011 6:45 pm

### Re: Can anyone help me finding the solution

Thank you for your reply. However I could not understand the solution you have given. The product of 32 and 26 is not 1664. Then how the answer can be correct? I could not find any two number less than 50 that produces 1664. If you can, please explain it in more easy way.

Sidharth
Posts: 20
Joined: Sat Feb 08, 2014 11:29 pm

### Re: Can anyone help me finding the solution

But the product of 32,26 and 2 is 1664

Labib
Posts: 411
Joined: Thu Dec 09, 2010 10:58 pm
Please Install $L^AT_EX$ fonts in your PC for better looking equations,