Problems on H.C.F and L.C.M
- 1. The least multiple of 7, which leaves a remainder of 4, when divided by 6, 9, 15 and 18 is?
Options- A. 74
- B. 94
- C. 184
- D. 364 Discuss
Correct Answer: 364
Explanation:
Let the least multiple of 7 be x, which when divided by 90 leaves the remainder 4. Then, x is of the form 90k + 4
Now, the minimum value of k for which 90k + 4 divisible by 7 is 4
? x = 90 x 4 + 4
= 364
- 2. The largest natural number, which exactly divides the products of any four consecutive natural numbers, is?
Options- A. 6
- B. 12
- C. 24
- D. 120 Discuss
Correct Answer: 24
Explanation:
In any 4 consecutive natural numbers:
Two numbers will be divisible by 2, one of which will be divisible by 4, at least one number will be divisible by 3.
These facts give us a lower bound of 2 x 4 x 3 = 24 for the solution.
But 24 is also the product of the first 4 consecutive natural numbers, so it is also an upper bound.
- 3. The total number of prime factors of the products (8) 20, (15) 24, (7) 15 is?
Options- A. 59
- B. 98
- C. 123
- D. 138 Discuss
Correct Answer: 123
Explanation:
Since 2, 3, 5, 17 are prime numbers and
The given expression = (23)20 x (3 x 5)24 x (17)15
= 260 x 324 x 524 x 1715
So the total number of prime factors in the given expression is (60 + 24 + 24 + 15 ) = 123.
- 4. The traffic lights at three different road crossings changes after every 48 sec., 72 sec. and 108 sec. respectively. If they all change simultaneously at 8 : 20 : 00 hrs, then they will again change simultaneously at?
Options- A. 8 : 27 : 12 hrs.
- B. 8 : 27 : 24 hrs.
- C. 8 : 27 : 36 hrs.
- D. 8 : 27 : 48 hrs. Discuss
Correct Answer: 8 : 27 : 12 hrs.
Explanation:
Interval of changes = (L . C . M of 48, 72, 108) sec. = 432 sec.
So, the lights will simultaneously changes after every 432 seconds i,e 7 min. 12 sec.
So, the next simultaneously changes will take place at 8 : 27 : 12 hrs.
- 5. Four metal rods of lengths 78 cm, 104 cm, 117 cm and 169 cm are to be cut into parts of equal length. Each part must be as long as possible. What is the maximum number of pieces that can be cut?
Options- A. 27
- B. 36
- C. 43
- D. 480 Discuss
Correct Answer: 36
Explanation:
Given, length of four metal roads are 78, 104, 117 and 169 cm.
Now,
78 = 13 x 2 x 3,
104 = 13 x 2 x 2 x 2,
117 = 13 x 3 x 3,
169 = 13 x 13
Length of each piece of rod as long as possible, HCF = 13 CM
? Number of piece = 6 + 8 + 9 + 13 = 36
- 6. There are five hobby clubs in a college viz. photography, yachting, chess, electronics and gardening. The gardening group meets every second day, the electronics group meets every third day, the chess group meets every fourth day, the yachting group meets every fifth day and the photography group meets every sixth day. How many times do all the five groups meets on the same day within 180 days?
Options- A. 3
- B. 5
- C. 10
- D. 18 Discuss
Correct Answer: 3
Explanation:
Gardening group meets once in 2 days, electronics group meets once in 3 days, chess group meets once in 4 days, yachting group meets once in 5 days and the photography group meets once in 6 days.
If they meet on the same day at one time, then the next time they will meet on same day again will be the LCM of 2, 3, 4, 5 and 6 which is equal to 60.
Hence, within 180 days all the five groups will meet on the same day = 180/60 = 3 times.
- 7. A, B and C start at the same time in the same direction to run around in 252s, B in 308s and C in 198s, all starting at the same point. After what time will they meet again at the starting point?
Options- A. 26 min and 18s
- B. 42 min and 36s
- C. 45 min
- D. 46 min and 12s Discuss
Correct Answer: 46 min and 12s
Explanation:
The time at which all the three persons meet will be the LCM of the time taken by each person individually to complete one round.
252 = 22 x 71 x 91
308 = 22 x 71 x 111
198 = 21 x 91 x 111
? LCM of 252, 308 and 198 = 22 x 71 x 91 x 111 = 2772
So, A, B and C will again meet at the starting point in 2772s i.e., 46 min and 12s
- 8. What is the LCM of (x 2 - y 2 - z 2 - 2yz), (x 2 - y 2 + z 2 + 2xz) and (x 2 + y 2 - z 2 - 2xy)?
Options- A. (x + y + z) (x + y - z) (x - y + z)
- B. (x + y + z) (x - y - z) (x - y + z)
- C. (x + y + z) (x + y - z) (x - y - z)
- D. (x + y - z) (x - y - z) (x - y + z) Discuss
Correct Answer: (x + y + z) (x - y - z) (x - y + z)
Explanation:
x2 - y2 - z2 - 2yz = x2 - (y + z)2 = (x+ y + z)(x - y - z)
x2 - y2 + z2 + 2yz = (x + z)2 - y2 = (x + y + z) (x + y - z)
x2 + y2 - z2 - 2xy = (x - y)2 - z2 = (x - y + z) (x - y - z)
? LCM = (x + y + z) (x - y - z) (x - y + z)
- 9. A man has four copper rods whose lengths are 52 m, 65 m, 78 m, 91 m, respectively. This man wants to cut pieces of same length from each of four rods. what is the least number of total pieces if he is to cut without any wastage?
Options- A. 23
- B. 24
- C. 22
- D. 20 Discuss
Correct Answer: 22
Explanation:
The pieces cut from four rods is least, so length of the pieces is the HCF of 52, 65, 78 and 91 which is 13.
? Number of pieces cut =(52 + 65 + 78 + 91)/13 = 22
- 10. A, B, and C start at the same time in the same direction to run around a circular stadium. A completes a round in 252s, B in 308s and C in 198s, all starting of the same point. After what time will they meet again at the string point?
Options- A. 26 min and 18s
- B. 42 min and 36s
- C. 45 min
- D. 46 min and 12s Discuss
Correct Answer: 46 min and 12s
Explanation:
The time which all the three persons meet will be the LCM of the time by each person individually to complete one round.
252 = 22 x 71 x 91
308 = 22 x 71 x 111
198 = 21 x 91 x 111
? LCM of 252, 308 and 198 = 22 x 71 x 91 x 111 = 2772
So,A,B and C will again meet at the starting point in 2772 i.e.,46 min and 12s.
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