The sum of two numbers is 60 . Each number is smaller than 40 and has exactly six
factors. Their common factors are 1,2 and 4 . What are these two numbers?

Question

The sum of two numbers is 60 . Each number is smaller than 40 and has exactly six factors. Their common factors are 1,2 and 4 . What are these two numbers?

UpStudy AI · Accepted Answer

Let the two numbers be $$a$$ and $$b$$ with

$$
a+b=60,
$$

and both $$a$$ and $$b$$ are less than 40 and have exactly six factors.

A number has exactly six factors if it is of one of the following forms:

1. $$p^5$$, where $$p$$ is a prime (giving $$5+1=6$$ divisors),
2. $$p^2q$$, where $$p$$ and $$q$$ are distinct primes (giving $$(2+1)(1+1)=6$$ divisors).

Since both numbers are less than 40, the candidates from these forms are limited.

1. The form $$p^5$$:
   - For $$p=2$$, we get $$2^5=32$$, which is less than 40.
   - For any larger prime, $$p^5$$ will exceed 40.

2. The form $$p^2q$$:
   - For $$p=2$$: 
     - $$2^2 \times 3=12$$,
     - $$2^2 \times 5=20$$,
     - $$2^2 \times 7=28$$.
   - For $$p=3$$:
     - $$3^2 \times 2=18$$.
   - (Any other choice yields numbers $$\ge40$$ or duplicates.)

Thus, the possible numbers with exactly six factors and less than 40 are:

$$
12,\; 18,\; 20,\; 28,\; 32.
$$

Next, we must choose two numbers from this list such that their sum is 60:

- Checking possible pairs, we find:

$$
28+32=60.
$$

Finally, we verify their common factors. The factor lists are:

- Factors of $$28$$: $$1,\,2,\,4,\,7,\,14,\,28$$,
- Factors of $$32$$: $$1,\,2,\,4,\,8,\,16,\,32$$.

Their common factors are:

$$
1,\; 2,\; 4.
$$

This meets the condition that the numbers have exactly the common factors $$1$$, $$2$$, and $$4$$.

Thus, the two numbers are $$28$$ and $$32$$.
The two numbers are 28 and 32.

The sum of two numbers is 60 . Each number is smaller than 40 and has exactly six
factors. Their common factors are 1,2 and 4 . What are these two numbers?

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Arithmetic Questions