HCF and LCM

[markosheehan]

John has 32 red balloons, 24 white balloons and 16 blue balloons. she wants to make a number of identical balloon arrangements for a party. what is the greatest number of arrangements she can make if all the balloons are used . how many balloons of each colour are in these arrangements.

[Guest]

She can make 8 identical arrangements at most. B=2, W=3, R=4.

[leesajohnson]

B=2, W=3, R=4.

[KelseyJohnson]

Variables and Multiples:

In the event that number a separated another number b precisely, we say that a will be a factor of b.

For this situation, b is known as a various of a.

Most elevated Common Factor (H.C.F.) or Greatest Common Measure (G.C.M.) or Greatest Common Divisor (G.C.D.):

The H.C.F. of at least two than two numbers is the best number that partitions every one of them precisely.

There are two techniques for finding the H.C.F. of a given arrangement of numbers:

Factorization Method: Express the every last one of the given numbers as the result of prime elements. The result of minimum forces of normal prime variables gives H.C.F.

Division Method: Suppose we need to discover the H.C.F. of two given numbers, separate the bigger by the littler one. Presently, isolate the divisor by the rest of. Rehash the way toward partitioning the previous number by the rest of acquired till zero is gotten as leftover portion. The last divisor is required H.C.F.

Finding the H.C.F. of in excess of two numbers: Suppose we need to discover the H.C.F. of three numbers, at that point, H.C.F. of [(H.C.F. of any two) and (the third number)] gives the H.C.F. of three given number.

Likewise, the H.C.F. of in excess of three numbers might be gotten.

Slightest Common Multiple (L.C.M.):

The slightest number which is precisely detachable by every single one of the given numbers is called their L.C.M.

There are two strategies for finding the L.C.M. of a given arrangement of numbers:

Factorization Method: Resolve every single one of the given numbers into a result of prime components. At that point, L.C.M. is the result of most astounding forces of the considerable number of variables.

Division Method (alternate way): Arrange the given numbers in a rwo in any request. Partition by a number which isolated precisely no less than two of the given numbers and convey forward the numbers which are not separable. Rehash the above procedure till no two of the numbers are distinguishable by a similar number with the exception of 1. The result of the divisors and the unified numbers is the required L.C.M. of the given numbers.

Result of two numbers = Product of their H.C.F. furthermore, L.C.M.

Co-primes: Two numbers are said to be co-primes if their H.C.F. is 1.

H.C.F. furthermore, L.C.M. of Fractions:

1. H.C.F. = H.C.F. of Numerators

L.C.M. of Denominators

2. L.C.M. = L.C.M. of Numerators

H.C.F. of Denominators

H.C.F. furthermore, L.C.M. of Decimal Fractions:

In a given numbers, make a similar number of decimal places by attaching zeros in a few numbers, if important. Considering these numbers without decimal point, discover H.C.F. or then again L.C.M. by and large. Presently, in the outcome, separate the same number of decimal places as are there in every one of the given numbers.

Correlation of Fractions:

Discover the L.C.M. of the denominators of the given parts. Change over every one of the parts into a proportionate portion with L.C.M as the denominator, by increasing both the numerator and denominator by a similar number. The resultant portion with the best numerator is the best.

[Guest]

I did this problem the old fashioned way....

I actually wrote out numbers that divided into 32, 24 and 16 (leaving no remainder) representing 32 red, 24 white and 16 blue balloons

These numbers are also called factors.

so...

Factors for 16 are 1, 2, 4, 8 and 16

Factors for 24 are 1, 2, 3, 4, 6, 8, 12 and 24

Factors for 32 are 1, 2, 4, 8, 16 and 32

From above the highest number of bundles using all the balloons are 8

Why?!?

Because 8 is the highest number that is common to all three numbers 18, 24 and 32

and this also gives a clue for the colour combination that is the same for each bundle.

So there are a total of 8 bundles.

16 blue balloons shared among the 8 bundles, each bundle will have 2 blue balloons

24 white balloons shared among the 8 bundles, each bundle will have 3 white balloons

32 red balloons shared among the 8 bundles, each bundle will have 4 red balloons.

The colour combination are 2 blues, 3 whites and 4 reds - a total of 8 balloons per bundle.

I won't tell you how long It had taken to work out the solution... it is embarrassing!!!!