Quiz – 1

L.C. of 3, 4, 7 10 and 12 = 2*2*3*5*7 = 420
In the above factorization, 3, 5 and 7 are not in pairs, we have to multiply 420 by 3*5*7
= 420*3*5*7
= 44100

for no values of n

HCF of ((138-63),(228-138),(228-63))
HCF of (75,90,165)
75=3*5*5
90=2*3*3*5
165=3*5*11
ans=3*5=15

Take LCM of 4,5,6,7. It is 420
BUt the no must leave remainder 2 in each case, so the no is of the form: 420k + 2.
The smallest 4-digit no is 1000. So keeping k=0,1,2,3....
We get that the largest no smaller than the smallest 4 -digit no is 842

Simply if divide all number with 5 than we get reminder as below :
--
90/5 = 18
80/5 = 16
70/5 = 14
60/5 = 12
50/5^2 = 2
40/5 = 8
30/5 = 6
20/5 = 4
10/5 = 2
--
So total highest power will be 9 and 1 add to form 50 terms
So total will be 10

If a − b is divisible by 3, then a − b = 3k, for some integer k
(a − b)² = (3k)²
a² − 2ab + b² = 9k²
a³ − b³ = (a−b) (a² + ab + b²)
= (a−b) (a² − 2ab + b² + 3ab)
= 3k (9k + 3ab)
= 3k * 3 (3k + ab)
= 9 k(3k+ab)
Since k(3k+ab) is an integer, then 9k(3k+ab) is divisible by 9

The question says how many powers of 5 are there in 30!
Only the numbers 5,10,15,20, 25, 30 are divisors of 5 and 25 is divisible by 5^2 ( 5*5)
Therefore the total number of five\'s are 7.
Hence the greatest positive power of 5 that divides 30! is 7.

166 * 6 + 5 = 1001 is the smallest 4-digit number gives a remainder of 5 when divided by 6.
So  N = 1001 + 6 * m   for some m.
so  N = 200 * 5 + 1 + 5 m + m
=  (200 + m) * 5 + 1+m
So Remainder when N is divided by 5 =   3  given
=>  remainder when 1+m is divided by 5  = 3
=>  remainder when  m is divided by 5  = 2
Smallest m is 2.
Hence, N = 1001 + 6 * 2 = 1013.

p=11k+7
so (p+4)(p+15)=(11k+11)(11K+22)
=11(k+1)*11(k+2)
=121(k+1)(k+2)
here one of the terms of (k+1)and(k+2)is even and another is odd..so multiplication of odd and even no. produce a even no..so the no. is divisible by (121*2)=242

LCM (3, 4, 5, 6 ,8) = 120
120 = 2 x 2 x 2 x 3 x 5.
As 2,3 and 5 are not in pair in LCM’s factor so we need to multiply 120 by 5 and 3,2 to make it a perfect square.
120 x 2 x 5 x 3 = 3,600.
∴ 3600 is the least perfect square divisible by 3, 4, 5, 6 and 8.