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SIXTH GRADE / GREATEST COMMON DIVISOR and LEAST COMMON MULTIPLE / TEST 3.1.1

MATHEMATICS IS EASY. PROVIDED THAT ONE WORKS ENOUGH.
ENOUGH IS SKILL DEPENDENT, HAVE TO FIND "ENOUGH" FOR YOUR SIZE.

SOME OF OUR QUESTIONS ARE REALLY HARD, FOR THOSE THAT ARE SKILLED IN MATH.
SKIP THEM IN CASE NOT FOR YOU.

IF YOU DESIRE SO, WE CAN DISCUSS HARD/INTERESTING QUESTIONS IN OUR FORUM.

 GCD abd LCM

    ♦ 
We preferred to introduce htese concepts earlier.
    ♦ We believed if a rational number is to be reduced, GCD is to be used.
    ♦ If the denominators are to be equal, the LCM is to be looked for.

    Ref.-1
    Ref.-2
1. Which choice is the prime factorized form of 324/126 ?

a) (2 x 2 x 3 x 3 x 3 x 3)/(2 x 3 x 3 x 7)
b) (2 x 3 x 3 x 3 x 3)/(2 x 3 x 3 x 7)
c) (2 x 2 x 3 x 3 x 3 x 3)/(2 x 3 x 7)
d) (2 x 2 x 3 x 3 x 3)/(2 x 3 x 3 x 7)
2,3,5,7,11,13, ... are "prime" numbers .
2. Which common (numerator-denominator) numbers reduces (2 x 2 x 3 x 3 x 3 x 3)/(2 x 3 x 3 x 7) ?

a) (
2 x 2 x 3 x 3 x 3 x 3)/ (3 x 3 x 3 x 7) i.e. 2 and 32
b) (
2 3 x 3 x 3 x 3 x 3 x 3 x 3)/ (2 x 3 x 3 x 7) i.e. 2 and 33
c) (
2 x 2 x 3 x 3 x 3 x 3)/ (2 x 3 x 3 x 7) i.e. 2 and 32
d) (
3 x 2 x 3 x 3 x 3 x 3)/(2 x 3 x 3 x 7) i.e. 2 and 33
Numbers common in num. and denom. may be dropped .
3. What is the intersection of sets of prime factors of 324 and 126 ?

a) {2 , 3, 3}
b) {2 , 3}
c) {1, 2 , 3}
d) ∅
In a set an element is not to repeat and 1 is not prime .
4. What may be the GCD algorithm deuduced from the previous 3 questions ?

a) the highest powers of common multiples are to be multiplied.
b) the lowest powers of common multiples are to be multiplied.
c) the lowest powers of common and uncommon multiples are to be multiplied.
d) the least powers of common prime factors are to be multiplied.
Common prime factors are elements of intersection of prime factor sets.
5. Which prime factorization algorithm may have been / is being used in this page ?

a) divisibilty by primes from the least prime 1 up is tested.
b) divisibilty by primes from the least prime 2 up is tested.
c) divisibilty by primes from the least prime 3 up is tested.
d) divisibilty is tested.
Divisibiliy by primes from 2 up is to be tested.
y
6. For 4th graders we proposed to obtain the prime factors of both numbers separately and multiplication of least powers of common prime factors .
!

Which choice may be a short cut during a test ?

In case of need for tips get the cursor on "y" or for help click at "ry1/2" .
RY1 RY2



a) to use a vertical line segment and to do prime factorization of all numbers at once.
b) not to use any line segment.
c) to use a vertical line segment for each number to be factorized.
d) to do what convenient is.


All primes that are common divisors, i.e. the least powers in prime factorizations, are reduced .
ClickN KIDS Beginning Reading & Spelling Programs Teacher Created Resources ClickN KIDS Beginning Reading & Spelling Programs
7. In a rectangular shaped garden with dimensions 96 m. x 45 m. trees are to be planted. 3 m. from each side is to be left free of trees. The distances between trees are to be same and max. possible for the case. How many trees may be planted ?

!


a) 4
b) 3
c) 7
d) 5
If we consider the free areas the dimensions are 90 m. by 39 m.
8. The weak part of a wall, having dimensions of 16 m. by 10 m. is to be reinforced by square shaped perforated plates. What is the minimal number of perforated plates necessary ?

!


a) 30
b) 35
c) 40
d) 38
To find the dimensions of the perforated plates is first due.
9. Barrels of volumes 75 l. , 90 l. and 120 l. having respectively must , vinegar and grape juice in, are poured into bottles of same volume. What may be the minumum number of bottles used ?

!


a) 75
b) 85
c) 83
d) 76
First find the GCD of these 3 numbers to find a proper bottle volume.
10. Given natural numbers A and B with A = (10xB + 72)/2xB . Which is the set of possible B values ?

a) {1,2,4,6,9,12,18}
b) {1,2,3,4,6,9,12,18}
c) {1,2,3,6,9,12,18}
d) {1,2,3,4,6,9,12}
A = 5 + 36/B should be valid. So possible B values are those dividing 36 without residue .
11. What is the relation between the divisors and prime factors, {2,3,5}, of the number 30 ?

!


a) 30 has 2 x 2 x 2 x 2 = 16 divisors.
b) 30 has 2 x 2 x 2 = 8 divisors.
c) 30 has 2 x 2 = 4 divisors.
d) 30 has 2 x 3 = 6 divisors
Divisors : 1 , 2, 3, 5, 6, 10, 15 and 30 .
y
12. The number 15 x 2N is known to have 12 divisors. What should N be ?

In case of need for tips get the cursor on "y" or for help click at "ry1/2".

RY1 RY2



a) 5
b) 1
c) 3
d) 2
The case of non repeating prime factors we have seen in the previous question. This question is to understand what otherwise happens.
We know that if N=1, the number 30 has 8 factors/divisors. This means N should be a pretty small number.
13. Which set is the set of common multiples of the numbers 24 and 64 ?

a) {192}
b) {192, 384, 576, ... }
c) {192, 386, 582, ... }
d) {194, 388, 584, ... }
The set of multiples of 24 = {24, 48, 72, 96, 120, 144, 168, 192, 216, 240, 264, 288, 312, 336, 360, 384, ...}
The set of multiples of 64 = {64, 128, 192, 256, 320, 384, ...} .
14. The prime factors of 24 and 64 respectively are 2,2,2,3 and 2,2,2,2,2,2 . How is the LCM to be attained ?

a)
2x2x2 is already included in 2x2x2x2x2x2x2x2 . We additionally get the noncommon 3 as another factor .
b)
2x2x2 is already included in 2x2x2x2x2x2x2 . We additionally get the noncommon 3 as another factor .
c)
2x2 is already included in 2x2x2x2x2x2x2x2. We additionally get the noncommon 3 as another factor .
d)
2x2x2 is already included in 2x2x2x2x2x2x2. We additionally get the noncommon 3 as another factor .


2x2x2 is already included in 2x2x2x2x2x2x2x2. So it is enough to use 2x2x2x2x2x2x2x2=64 and 3 which is not a common factor.
15. What can be deduced from the previous 3 questions on obtaining the LCM ?

a) The smallest power of common prime factors and all of non-commons are to be multiplied .
b) All non-common prime factors are to be multiplied.
c) The greatest power of common prime factors are to be multiplied.
d) The greatest power of common prime factors and all of non-commons are to be multiplied.
The greatest power of common prime factors and all of non-commons are to be multiplied .
16. Prime factorization of the numbers 324 and 126 are given in the figure. What is the LCM ?

!


a) 32x34x7 = 15 876
b) 22x34x7 = 15 876
c) 22x34x7 = 2 268
d) 32x34x7 = 2 268
The greatest power of common prime factors and all of non-commons are to be multiplied .
17. The set of common primes of 2 prime factorized numbers and the set of powers of these are given respectively as C = {2,3,5,11} and P = {2,2,1,1}. What is the LCM ?

a) 2 012
b) 1 980
c) 1 998
d) none
The non-common factors are also needed .
y
18. For 4th graders, to calculata LCM, we proposed to obtain the prime factors of both numbers separately and multiplication of greatest powers of all common prime factors and the non-common factors.
!

Which choice may be a short cut during a test ?


In case of need for tips get the cursor on "y" or for help click at "ry1/2" .
RY1 RY2


a) to use a vertical line segment and to do prime factorization of all numbers at once.
b) not to use any line segment at all.
c) to use a vertical line segment for each number to be factorized.
d) to do what convenient is.
It takes less time to use only one vertical line.
19. The master has forgotten the window and the building owner has ordered him a perfectly square shaped window. What will be the minimal number of bricks to be broken if brick dimensions are 6 cm. by 15 cm. ?

!

a) 8
b) 18
c) 10
d) 22
Since LCM is 30,the square must have dimensions of 30 cm. by 30 cm.
20. Whoever climbs the stairs shown in the figure, steps down on the stair just before the last if he climbs 2 by 2, 3 by 3 and 5 by 5. How many stairs should there be ?

!

a) 31
b) 28
c) 35
d) 43
Finf LCM of 2,3 and 5 and add 1 .
21. A rectangular prism made up of bricks having dimensions 8, 9 and 10 cm. is to be placed in a sphere. How many brick will the prism have if it is supposed to have max. number of bricks in its volume ?

!


a) 32 400
b) 4 800
c) 64 800
d) 5 800
A cube is a special rectangular prism having max volume for a given total side length. This prism shold be a cube which means we have to find the LCM of 8,9 and 10.Then ...
y
22. Which is the minimal number that is divisible by 3, 5, and and 11 with a residue of 1 and by 13 without residue ?

In case of need for tips get the cursor on "y" or for help click at "ry1/2" .

RY1 RY2



a) 15 028
b) 10 044
c) 12 066
d) 12 128


3,5,7, and 11 are primes and their LCM is the product of all 4 .
23. What is the minimal value of a if 15687-a is divisible by 3, 4, 5, 6, and 9 ?

a) 25
b) 33
c) 27
d) 17
15 687 / 180 = 87.15 ve 87 x 180 = 15 660
24. Barrels of volumes 60, 84 and 96 liters containing must, vinegar and grape juice respectively are to be placed all into 96 l. barrels. How many will be the minimal number of barrels if no mixing and rest is permitted ?

!


a) 15
b) 9
c) 11
d) 13
LCM of 60 and 96 is 60 , 96 | 25 , 3 , 5
LCM of 84 and 96 is 84 , 96 | 25 , 3 and 7 .
1 96 l. barrel is to be used. For the others LCMs of 60 and 96 and LCM of 84 and 96 are needed. Then ... .
25. How many element does the set { A | 1000 < A < 2000 , natural number divisible by both 36 and 84 } have ?

a) 2
b) 4
c) 1
d) 8
FinfdLCM of 36 and 84 .
This is the feedback!

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