Practice Quantitative Aptitude questions on HCF and LCM, covering highest common factor, least common multiple, prime factorization, divisibility, remainders, and word problems involving repeated events, grouping, and equal distribution. Takes about 20–25 minutes and is useful for placement tests, TCS NQT, SSC, banking and MBA entrance exams.
Question 1
The length and breadth of a rectangular floor are 16.25 metre and 12.75 metre respectively. Find how many minimum number of square tiles would be required to cover it completely?
Explanation
Convert to cm: 1625 cm x 1275 cm. The side of the largest square tile = HCF(1625, 1275) = 25 cm.
Number of tiles = 162525 × 127525 = 65 × 51 = 3315.
Number of tiles = 162525 × 127525 = 65 × 51 = 3315.
Question 2
LCM of two numbers is 138. But their GCD is 23. The numbers are in a ratio 1:6. Which is the largest number amongst the two?
Explanation
GCD = 23, so the numbers are 23a and 23b with the ratio 1 : 6, that is 23 and 138.
Check: the LCM of 23 and 138 = 138. The larger number = 138.
Check: the LCM of 23 and 138 = 138. The larger number = 138.
Question 3
If least common multiple of two numbers is 225 and the highest common factor is 5 then find the numbers when one of the numbers is 25?
Explanation
LCM × HCF = the product of the two numbers.
225 × 5 = 25 × the other number, so the other number = 112525 = 45.
225 × 5 = 25 × the other number, so the other number = 112525 = 45.
Question 4
The greatest number of four digits which is divisible by 15, 25, 40, 75 is
Explanation
The LCM of 15, 25, 40 and 75 = 600.
The greatest 4-digit number is 9999. 9999 ÷ 600 gives the remainder 399.
The required number = 9999 - 399 = 9600.
The greatest 4-digit number is 9999. 9999 ÷ 600 gives the remainder 399.
The required number = 9999 - 399 = 9600.
Question 5
The sum of two numbers is 156 and their HCF is 13. The numbers of such number pairs is
Explanation
Let the numbers be 13a and 13b, where a and b are co-prime. 13(a + b) = 156, so a + b = 12.
The co-prime pairs with a sum of 12 are (1, 11) and (5, 7). The pairs (2, 10), (3, 9), (4, 8) and (6, 6) are not co-prime.
So there are 2 such pairs.
The co-prime pairs with a sum of 12 are (1, 11) and (5, 7). The pairs (2, 10), (3, 9), (4, 8) and (6, 6) are not co-prime.
So there are 2 such pairs.
Question 6
What will be least number of marbles with Rohit if he can arrange them in rows of 18, 10 and 15 marbles each as well as make a perfect solid square out of all the marbles?
Explanation
The number of marbles is a multiple of 18, 10 and 15, so it is a multiple of their LCM = 90 = 2 × 3² × 5.
To make a perfect square, every prime must have an even power, so multiply by 2 × 5 = 10: 90 × 10 = 900 = 30².
The least number of marbles = 900.
To make a perfect square, every prime must have an even power, so multiply by 2 × 5 = 10: 90 × 10 = 900 = 30².
The least number of marbles = 900.
Question 7
What least possible 4-digit number, when divided by 12, 16, 18 and 20 leaves 21 as remainder?
Explanation
The number leaves the remainder 21 when divided by 12, 16, 18 and 20, so it is of the form LCM × k + 21.
The LCM of 12, 16, 18 and 20 = 720. The 4-digit multiples of 720 start at 1440, so the least number = 1440 + 21 = 1461.
The LCM of 12, 16, 18 and 20 = 720. The 4-digit multiples of 720 start at 1440, so the least number = 1440 + 21 = 1461.
Question 8
There are 3 equilateral triangles with sides 114cm, 76cm and 152 cm. What maximum size scale can measure them exactly?
Explanation
The scale must measure 114 cm, 76 cm and 152 cm exactly, so its length is the HCF of these numbers.
114 = 2 × 3 × 19, 76 = 2² × 19 and 152 = 2³ × 19, so the HCF = 38 cm.
114 = 2 × 3 × 19, 76 = 2² × 19 and 152 = 2³ × 19, so the HCF = 38 cm.
Question 9
What largest number will divide 47, 35 and 27 leaving same remainder in each case? What will be the common remainder?
Explanation
The number dividing 47, 35 and 27 with the same remainder divides their differences: 47 - 35 = 12, 35 - 27 = 8 and 47 - 27 = 20.
The HCF of 12, 8 and 20 = 4.
The remainder = 47 ÷ 4 leaves 3, so the number is 4 and the common remainder is 3.
The HCF of 12, 8 and 20 = 4.
The remainder = 47 ÷ 4 leaves 3, so the number is 4 and the common remainder is 3.
Question 10
HCF and LCM of two numbers is 8 and 96. Sum of those numbers is 56. Then what is sum of their reciprocals?
Explanation
The product of the two numbers = HCF × LCM = 8 × 96 = 768. Their sum = 56.
The sum of their reciprocals = sumproduct = 56768 = 796.
The sum of their reciprocals = sumproduct = 56768 = 796.
Question 11
Manoj wants to paste wallpaper on wall of his room. The wall is 4 meters and 50 cm in length and 3 meters and 50 cm in height. But the wall should be covered completely only by square pieces of wallpaper having same size. What is the number of maximum sized wallpaper squares needed to cover the walls completely?
Explanation
The wall is 450 cm × 350 cm. The largest square piece that covers it completely has a side equal to the HCF of 450 and 350, that is 50 cm.
Number of squares = 45050 × 35050 = 9 × 7 = 63.
Number of squares = 45050 × 35050 = 9 × 7 = 63.
Question 12
The L.C.M. Of 2, 2.7 and 0.09 is:
Explanation
Convert the numbers to whole numbers: 2 = 200, 2.7 = 270 and 0.09 = 9 (in hundredths).
LCM of 200, 270 and 9 = 5400.
Converting back, the LCM = 5400100 = 54.
LCM of 200, 270 and 9 = 5400.
Converting back, the LCM = 5400100 = 54.
Question 13
A man decides to pave with square tiles his hall which is 4.8 metres long and 5.38 metres wide. Find the largest size of the tile that he could use.
Explanation
The tiles must fit the length and the width exactly, so the largest tile side is the HCF of 480 cm and 538 cm.
480 = 2⁵ × 3 × 5 and 538 = 2 × 269, so the HCF = 2 cm.
480 = 2⁵ × 3 × 5 and 538 = 2 × 269, so the HCF = 2 cm.
Question 14
The least number which is a multiple of 11 and when divided by 3, 5 and 9 leaves 2 as remainder:
Explanation
The number leaves the remainder 2 when divided by 3, 5 and 9, so it is of the form 45k + 2 (LCM of 3, 5, 9 = 45).
It must be a multiple of 11: k = 1 gives 47, 2 gives 92, 3 gives 137, 4 gives 182, 5 gives 227, 6 gives 272, 7 gives 317, 8 gives 362 and 9 gives 407 = 11 × 37.
So the least such number is 407.
It must be a multiple of 11: k = 1 gives 47, 2 gives 92, 3 gives 137, 4 gives 182, 5 gives 227, 6 gives 272, 7 gives 317, 8 gives 362 and 9 gives 407 = 11 × 37.
So the least such number is 407.
Question 15
What is the smallest number which is a perfect square and is divisible by 16, 20 and 24?
Explanation
The LCM of 16, 20 and 24 = 240 = 2⁴ × 3 × 5.
For a perfect square every prime must appear an even number of times, so multiply by 3 × 5 = 15: 240 × 15 = 3600.
3600 = 60², so the smallest such number is 3600.
For a perfect square every prime must appear an even number of times, so multiply by 3 × 5 = 15: 240 × 15 = 3600.
3600 = 60², so the smallest such number is 3600.
Question 16
When 5, 8 and 12 divide a multiple of 13, they leave remainders of 3, 6 and 10 respectively. Find the least such number.
Explanation
Since 5-3=8-6=12-10=2, N+2 must be a multiple of LCM(5,8,12)=120, i.e. N=120j-2. Also N must be a multiple of 13: solving 120j-2 is congruent to 0 mod 13 gives j=5, so N=598.
Question 17
Find two natural numbers, whose sum is 85 and whose LCM is 102.
Explanation
With GCD d=17, numbers are 17m and 17n where mn=6, m+n=5, and gcd(m,n)=1: m=2,n=3 works, giving 34 and 51.
Question 18
What is the HCF of 143 and 208?
Explanation
143 = 11 × 13 and 208 = 24 × 13. The common factor is 13.
Question 19
Which is the least number greater than 12, which, when divided by 12 and 16, leaves the same remainder in each case, but is exactly divisible by 11?
Explanation
The multiples of 11 above 12 are 22, 33, 44, 55.
22 leaves remainders 10 and 6, 33 leaves 9 and 1, and 44 leaves 8 and 12 on division by 12 and 16, so none of them has equal remainders.
55 leaves 7 on division by 12 and 7 on division by 16, and 55 / 11 = 5 exactly. So the least number is 55.
22 leaves remainders 10 and 6, 33 leaves 9 and 1, and 44 leaves 8 and 12 on division by 12 and 16, so none of them has equal remainders.
55 leaves 7 on division by 12 and 7 on division by 16, and 55 / 11 = 5 exactly. So the least number is 55.
Question 20
The H.C.F. of two numbers is 11 and their L.C.M. is 7700. One number is 275. Find the other one.
Explanation
Product of numbers = HCF x LCM = 11 × 7700 = 84700. Other number = 84700275 = 308.
Question 21
Find the LCM of 24, 54 and 70.
Explanation
24 = 23 × 3, 54 = 2 × 33, 70 = 2 × 5 × 7. LCM = 23 × 33 × 5 × 7 = 7560.
Question 22
Packets of chocolates are kept in a box in the form of a cube. If the total number of packets is 2197, find the number of packets in a row of each layer.
Explanation
2197 = 133, so the cube has 13 packets along each edge, giving 13 packets in a row of each layer.
Question 23
Find the greatest number of 5 digits which when divided by 8, 12, 15 and 20 leaves respectively 5, 9, 12 and 17 as remainders.
Explanation
Each divisor minus its remainder is 3 (8-5=3, 12-9=3, 15-12=3, 20-17=3). LCM(8,12,15,20)=120. The number is of the form 120m-3. The largest 5-digit value is found from (99999+3)/120=833.35, so m=833 gives 120 × 833-3 = 99957.
Question 24
Find the least number which when divided by 16, 12, 18 and 30 leaves remainder 4 in each case but is completely divisible by 7.
Explanation
LCM(16,12,18,30) = 720. The number is of the form 720k+4. Testing k=4 gives 2884, which is exactly divisible by 7 (2884 = 7 × 412).
Question 25
Find the highest number of 4 digits which is completely divisible by 48, 40 and 64.
Explanation
LCM(48, 40, 64) = 960. The largest 4-digit multiple of 960 is 960 × 10 = 9600.
