# Perfect and amicable numbers

1) See attached

2) Show that 28 is a perfect number.

3) What is the amicable number to 17296?

4) Express all 5/11 as the sum of distinct unit fractions.

5) Using modular arithmetic, derive a formula to find all numbers that meet the following conditions:

- Divide by 3 the remainder is 1

- Divide by 5 the remainder is 0

6) What is the area of a triangle with lengths of sides 28, 30 and 32?

7) What is the probability of picking 6 numbers out of 53 numbers and getting

4 out of 6 numbers correct?

8) Lagrange contributed a great deal to the metric system. How would you show your students how to use the metric highway below?

See attached

9) See attached

10) See attached

11) Chevalier de Mere's Problem-

How many times must you throw two die in order to have half a chance of getting double sixes? (The original problem used the word dice but die is singular and dice is plural i.e., a pair of dice is two die.)

We know that (see attached). If you roll dice you have 36 outcomes. Also the probability of getting at least one double 6 on n rolls of the dice is the same as getting no double sixes on n rolls of a dice.

The probability of double sixes is 1/36. The probability of any combination other than double sixes is 35/36.

Thus, Pascal and Fermat concluded that solving for n in the formula below will solve the problem.

See attached

What is the minimum number of rolls of the dice to have at least 1/2 chance of getting double sixes?

12) Explain how to multiply 88 times 112 using Egyptian multiplication.

13) What 3 expressions can generate all Pythagorean triples.

14) Find an arithmetical progression with 5 terms, sum 11, and common difference 1/2.

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#### Solution Preview

Hi there,

Thanks for letting me work on your post. I've included my explanations in the word document.

1) valuate .

Sum=(7+100)*(100-7+1)/2=5029

2) Show that 28 is a perfect number.

A perfect number could have the expression of 2^(p-1)*(2^p-1) in which p and 2^p-1 are both Mersenne prime. For 28, p=3 and both p and 2^3-1 (7) are Mersenne prime.

3) What is the amicable number to 17296?

aliquot divisors of number 17296 are 1, 2, 4, 8,16,23,47,368,752,1081,2162,4324,8648

therefore, amicable number: 1+2+4+8+16+23+46+47+92+94+184+188+368+376+752+1081+2162++4324+8648

=18416

4) Express all as the sum of distinct unit fractions.

5/11-1/3=4/33

4/33-1/9=1/99

Therefore, 5/11=1/3+1/9+1/99

5) Using modular arithmetic, derive a formula to find all numbers that meet the following conditions:

• Divide by 3 the remainder is 1

Since 4 has remainder of 1 when it is divided by 3 (4≡1(mod 3)),

4+3n (n is any integer) is the number for which the remainder is 1 when it is divided by ...

#### Solution Summary

Perfect and amicable numbers are provided. Expressions to generate all Pythagorean triples are determined.