Problems on HCF and LCM - Aptitude Questions Part 3

11. Given:
Eq 1: x2 - 8x + 15
Eq 2: x2 - px +1
HCF of Eq1 & Eq 2 = (x - 4)
Find the value of p.

a.8
15
b.15
8
c. 4
d. 5

Answer: c. 4

Explanation:

(x - 4) is HCF i.e. a factor of both equations
∴ The equations must get satisfied for x = 4
Also, when x = 4, both equations are equal in value.
Putting x = 4 in both equations
42 - 8(4) + 15 = 42 - p(4) +1
∴ 31 - 32 = 17 - 4p
∴ p = 4


12. The given five signals light up automatically at intervals of 3 min, 4 minutes, 8 min, 10 min, and 12 min respectively. How many times in 8 hours will they light up together from the time they start?

a. 3 times
b. 4 times
c. 5 times
d. 12 times

Answer: b. 4 times

Explanation:

We need the next instances when the signals light up.
That means the Least Common Multiple (LCM) of 3, 4, 8, 10, 12
3, 4 divide 12 so neglect them.
LCM of 8, 10 and 12
-----------------------------------
4           8           10           12
-----------------------------------
2           2           10           3
             1           5             3
-----------------------------------
∴ LCM = 4 x 2 x 5 x 3 = 120 = They light up together after 2 hours

After starting, they light up together 1st time in 2 hours.
Then 2nd time in 2 + 2 = 4 hours.
Then 3rd time in 6 hours.
And 4th time in 8 hours.


13. In a race on a circular track, the three athletes complete one round in 27 minutes, 45 minutes and 63 minutes respectively. Find the time after which they meet again at the starting point, since the time they started.  

a. 9 hours
b. 126 minutes
c. 135 minutes
d. 945 minutes

Answer: d. 945 minutes

Explanation:

We need the next instance that means the LCM of times of all 3 athletes.
--------------------------------------------
9           27           45           63
--------------------------------------------
             3             5              7
-------------------------------------------
∴ LCM = 9 x 3 x 5 x 7 = 945 = They meet after 945 minutes


14. A child goes to play with some pebbles in his bag. The number of pebbles is such that he can arrange them in rows of 18, 10 and 15 & form a square in each case. How many minimum number of pebbles are there in his bag?

a. 43
b. 90
c. 133
d. 900

Answer: d. 900

Explanation:

We need the least number, which means we must first find LCM of 18, 10 and 15
-------------------------------------------------
3           18           10           15
---------------------------------------------
2           6             10            5
5           3             5              5
             3             1              1
-----------------------------------------
∴ LCM = 3 x 2 x 5 x 3 = 90

90 is not a perfect square.
90 = 3 x 3 x 2 x 5
Here there are two 3's (this gives square of 3) but only one 2 and one 5
If we multiply by 2 and 5 then we get, 3 x 3 x 2 x 2 x 5 x 5 = 900
900 is perfect square = number of marbles


15. A wall is 4.5 meters long and 3.5 meters high. Find the number of maximum sized wallpaper squares, if the wall has to be covered with only the square wall paper pieces of same size.

a. 8
b. 12
c. 15.75
d. 63

Answer: d. 63

Explanation:

Wall can be covered only by using square sized wallpaper pieces.
Different sized squares are not allowed.
Length = 4.5 m = 450 cm;
Height = 3.5 m = 350 cm
Maximum square size possible means HCF of 350 and 450
We can see that 350 and 450 can be divided by 50.
On dividing by 50, we get 7 and 9.
Since we cannot divide further, HCF = 50 = size of side of square

Number of squares =Wall area=450 x 350= 63
Square area50 x 50

▲