Q16. Find the least number which when divided by 20, 25, 35, and 40 leaves remainders 14, 19, 29, 34 respectively.
A. (20 – 14) = 6
(25 – 19) = 6
(35 – 29) = 6
(40 – 34) = 6
So, required number = LCM{20,25,35,40} – 6
required number = 1400 – 6 => 1394.
Q17. Find the least number which when divided by 5, 6, 7, and 8 leaves a remainder 3, but when divided by 9 leaves no remainder.
A. LCM{5,6,7,8} = 840
So, it is quite evident that number should be of form : 840x + 3.
Also, it is given that (840x +3) % 9 = 0
So, least number for x = 2
So, required number = 840 * 2 + 3 = 1683.
Q18. The product of two numbers is 2028 and their HCF is 13. The number of such pairs is :
A. Since HCF is 13, so numbers will be 13x and 13y.
Also, (13x)(13y) = 2028
xy = 12
Now, pairs with product 12 : (1,12), (3,4).
Q19. HCF of two numbers is 11 and their LCM is 7700. If one of the numbers is 275, then the other is :
A. We know the formula :
275 * n2 = 11 * 7700
n2 = 308.
Q20. The greatest number which on dividing 1657 and 2037 leaves remainders 6 and 5 respectively, is :
A. Required Number : HCF{1657-6, 2037-5}
=> 127.