PRIMITIVE PYTHAGOREAN TRIPLES

Every primitive Pythagorean triple (a, b, c) is of the form a = m^2 − n^2, b= 2mn, c = m^2 + n^2. If m is even then n must be odd or the other way round.

Some basic properties of Pythagorean triples:
1. Exactly one leg is even.
2. Exactly one leg is divisible by 3.
3. Exactly one side is divisible by 5.
4. The area is divisible by 6.
5. The area of a Pythagorean triangle can never be a square number. Indeed, there is no Pythagorean triangle with two sides whose lengths are square
numbers.

MATHEMATICS BRAIN TEASER

HOW MANY CAMPERS?

Kem Bestari's cook, Puan Asiah, was just about to begin preparing the picnic lunch for all the campers. She already knew she needed to fill 55 bowls of the same size and capacity with the same amount of food. When she was done, she decided to read the guidelines for the picnic, just out of curiosity.

The guidelines said:
1. Every camper gets their own bowl of soup.
2. Every two campers will get one bowl of mee to share.
3. Every three campers will get one bowl of salad to share.
4. All campers are required to have their own helping of salad, mee, and soup.

After some rapid calculations, Puan Asiah was able to figure out how many campers were going to the picnic. Can you?

GOOD LUCK
Submit your solution to your Maths's teacher...

RIDDLE

Five Of A Kind

The first is needed to make quotes you see,
And it often sticks up when it's time for noon tea.

The second's biggest distinction is found
Bearing the symbol of love that is bound.

The third should be biggest but that can depend,
Never standing alone or it may offend.

The fourth is oft used when making a selection
Or if you should need a gun for protection.

The fifth is the fattest and oddest by far,
And can sometimes be found in a wrestling war.
What are they?

MATHS GIGGLES TABLE 1 TEACHER : Why are you doing your math sums on the floor? STUDENT : You told me to do it without using tables! TABLE 2 TEACHER : Please refer to the table. Circle number two and all the multiples of two. STUDENT : Teacher I can't see anything on my table.

PYTHAGOREAN TRIPLE It is surprising that there are some right-angled triangles where all three sides are whole numbers. The three whole number side-lengths are called a Pythagorean triple. An example is a = 3, b = 4 and c = 5, called "the 3-4-5 triangle". 32+42 = 9 + 16 = 25 = 52 so a2 + b2 = c2. Notice that the greatest common divisor of the three numbers 3, 4, and 5 is 1. Pythagorean triples with this property are called primitive. From primitive Pythagorean triples, you can get other, imprimitive ones, by multiplying each of a, b, and c by any positive whole number d > 1. This is because a2 + b2 = c2 if and only if (da)2 + (db)2 = (dc)2. Thus (a,b,c) is a Pythagorean triple if and only examples if (da,db,dc) is. For example, (6,8,10) and (9,12,15) are imprimitive Pythagorean triples.

Here are the of the first few primitive Pythagorean triples a b c a b c a b c 3 4 5 5 12 13 7 24 25 8 15 17 9 40 41 11 60 61 12 35 37 13 84 85 16 63 65 36 77 85 39 80 89 48 55 73

CHALLENGE YOURSELF

1. Find the only Pythagorean triangles with an area equal to their perimeter. 2. Ahmad has a triangular orchard with sides measuring 560 m, 420 m and 300 m. Determine whether the shape of his orchard is a right-angled, obtuse-angled or acute-angled triangle.