Factors and Multiples
If a number P divides another number Q exactly, we say that P is a factor of Q i.e. Q is a multiple of P.
H.C.F
The H.C.F of two or more than two numbers is the greatest number that divides each of them exactly.
L.C.M
The least number which is exactly divisible by each one of the given numbers is called their L.C.M
Product of two numbers
Product of their H.C.F and L.C.M
Co-primes
Two numbers are co-primes if their H.C.F is 1.
H.C.F and L.C.M of fractions
- H.C.F = H.C.F of Numerators⁄L.C.M of Denominators
- L.C.M = L.C.M of Numerators⁄H.C.F of Denominators
Solved Examples
Answer - B
Explanation
Prime numbers which are common to all the given numbers are 2,5 ,7. ∴ H.C.F = (22*5*73)= (4*5*343) = 6860
Answer - A
Explanation
108 = (22*33) , 360 = (23*32*5) and 600 = (23*52*3) ∴ H.C.F = (22* 3) = (4* 3)=12
Answer - A
Explanation
Remainder of 185/148 = 37 Remainder of 148/37 = 0 ∴ H.C.F. = 37
Answer - B
Explanation
Remainder of 1190/204 = 170 Remainder of 204/170 = 34 Remainder of 170/34 = 0 ∴ H.C.F. of 204, 1190 = 34 Remainder of 1145/34 = 17 Remainder of 34/17 = 0 ∴ H.C.F. of 204, 1190 and 1145 = 17
Answer - C
Explanation
First we find the H.C.F of 391 and 667. Remainder of 667/391 = 276 Remainder of 391/276 =115 Remainder of 276/115 = 46 Remainder of 115/46 = 23 Remainder of 46/23= 0 ∴ H.C.F. of 391, 667 = 23 ∴ 391/667 =( 391/23)/ (667/23)= 17/29
Q 6 - Find the L.C.M of (22*32*5*7) , (2³*3*52*72) and (2*3*7*11).
Answer - A
Explanation
We have L.C.M = product of terms containing highest powers of (2,3,5,7,11) = (2³* 32* 52*72*11) = (8*9*25*11*49)= 970200
Answer - D
Explanation
15 = 3 * 5 18 = 2* 3 * 3 = 2 * 3 2 24 = 2 *2 * 2 * 3 = 2 3 * 3 27 = 3 * 3 * 3 = 3 3 56 = 2 * 2 * 2 * 7= 2 3 * 7 L.C.M = product of terms containing highest powers of (2,3,5,7) =2 3 * 3 3 * 5 * 7 = 7560
Answer - D
Explanation
H.C.F of 2,8,10,16 = 2 L.C.M of 3,9,27,81 = 81 H.C.f = H.C.F of 2,8,10,16/L.C.M of 3,9,27,81 = 2/81 L.C.M = L.C.M of 2,8,10,16/H.C.F of 3,9, 27,81 = 80/3
Q 9 - Two numbers are in the ratio 8:11 . Considering their H.C.f as 6, find the numbers.
Answer - B
Explanation
Let the numbers be 8x and 11 x. then, their H.C.F = x So, the numbers are (8*6), (11*6) i.e 48 and 66.
Q 10 - Given the H.C. F of two numbers as 7 and their L.C.M as 210. If one of the numbers is 35, find the other.
Answer - B
Explanation
Let the Other number be X. then, Product of numbers = product of their H.C .F and L.C.M 35*x= 7* 210 ⇒ x= 7*210/35 = 42 Hence, the other number is 42.
Q 11 - Three big drums contain 36 liters, 45 liters and 72 liters of oil. What is the biggest measure which can measure all the different quantities exactly?
Answer - A
Explanation
Required measure = H.C.F of 36 L, 45 L, and 72 L = (32) liters = 9 liters [As 36 = 22*32, 45 = 32*5 and 72 = 24* 34]
Q 12 - Four electronic devices make a beep after duration of 30 minutes, 1 hour, 3/2 hours and 1 hour 45 min. respectively. If all the devices beeped together at 12 noon at what time will they beep together again?
Answer - A
Explanation
Intervals of beeping 30 min, 60 min, 90 min, 105 min. Interval of beeping together= L.C.M of 30 min. 60 min. 90 min. 105 min = (3*5*2*2*3*7) min. = 1260 min = 21 hrs. So, they will beep together again next morning at 9 am.
Answer - B
Explanation
Remainder of 783/513 = 270 Remainder of 513/270 =243 Remainder of 270/243 = 27 Remainder of 243/27 = 0 Remainder of 46/23= 0 ∴ H.C.F. of 513, 783 = 23 Remainder of 1107/23 = 0 ∴ H.C.F. of 513, 783 and 1107= 23
Q 14 - Find the smallest number which is exactly divisible by each one of the numbers 12, 15, 20 and 27.
Answer - A
Explanation
Required no. = L.C.M of 12,15, 20 and 27 = (3*2*2*5*9) = 540
Q 15 - Find the least number which if divided by 6, 7, 8, 9, 12 leaves the same remainder 2 in each case.
Answer - A
Explanation
Required number = (L.C.M of 6,7,8,9,12)+2 = (2*3*2*7*2*3)+ 2 = (504+2)= 506.
Q 16 - Find the largest natural number which can divide the product of any 4 consecutive natural numbers.
Answer - B
Explanation
(1*2*3*4) = 24 ∴ Required number = 24
Q 17 - Find the least number which if divided by 35, 45 and 55 leaves the remainder 18, 28 and 38 respectively.
Answer - A
Explanation
Here (35-18) = 17 , (45-28)= 17 and (55- 38) = 17 Required number = (L.C.M of 35,45, 55)- 17 = (3465 -17) = 3448
Answer - A
Explanation
H.C.F = H.C.F of 1,2,3,4/ L.C.M of 2,3,4,5 = 1/120
Answer - C
Explanation
H.C.F = H.C.F of 2, 8,10, 32/ L.C.M of 3,9, 27, 81 = 2/81
Q 20 - Which of the following is a pair of Co-primes?
Answer - B
Explanation
H.C.F of 18 and 25 is 1.
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