Mathematics I
Primary Education
Year 2015-2016
Practice 1.3 (October 6)1
You have to fill this poll http://goo.gl/forms/0aZ7bQDrc4 before Monday 5, 10 pm.
1. Show that the addition of three consecutive multiples of 17 is a multiple of 3.
2. How many squares are there in the next figure of the series? And in the 10th? In the number
n?
1
2
3
4
3. How many cubes are there in the 5th figure in the series? And in the 10th?
4. Find an expression for the area of the shaded region.
c Is 667 a prime number? And 673?
5. 667 = 23 · 29.
673 es primo
6. Find all prime number bigger than 200 and smaller than 230.
211, 223, 227, 229
7. Find three examples of numbers with an odd number of divisors. Can you see a property that
have all numbers that have an odd number of divisors?
1
c can be solved using a calculator.
Problems marked with the symbol 8. How many divisors does 2376 have? How many of them are odd? Find all divisors of 2376
that are multiple of 12.
2376 = 23 × 33 × 11
c Given a natural number n, consider the number an = n(n − 1) + 41. Is an always a prime
9. number?
10. Find three numbers that have 6 divisors. What is the smallest number that has 6 divisors?
Repite the problem for numbers with 12 divisors.
11. In a high school there are 100 lockers, with numbers from 1 to 100, and there are also 100
students numbered from 1 to 100. Student number 1 goes and opens all lockers. Next, student
number 2 goes in and closes all lockers with even numbers. In the general step, student number
k changes the state (opens the locker if it was closed and closes it if it was open) of all lockers
labeled with a multiple of k (and ignores the rest). Which lockers will be open after student
number 100 does that?
Additional problems
1. Show that the addtion of three consecutive multiples of 4 is a multiple of 12.
2. Find a number bigger than 250 and smaller than 300 that has 8 divisors. (January 2015)
Sol: 23 · 37 = 296
3. The number of points in the figures are called pentagonal numbers. The 2nd pentagonal
number, P2 , is 5. Can you find the 5th pentagonal number, P5 ? And the 10th one, P10 ? And
the n-th one, Pn ?
P1
P2
P3
P4
Sol: P10 = 145. Pn is more difficult. Don’t worry too much if you don’t get if. Pn =
n(3n − 1)
2