Tech Mahindra Divisibility Quiz 1

L.C. M of 3, 4, 7 10 and 12 = 2 x 2 x 3 x 5 x 7 = 420
In the above factorization, 3, 5 and 7 are not in pairs, we have to multiply 420 by 3 x5 x 7
= 420 x 3 x 5 x 7
= 44100

for no values of n

HCF of ((138-63),(228-138),(228-63))
HCF of (75,90,165)
75=3 x 5 x 5
90=2 x3 x 3 x 5
165=3 x 5 x 11
ans=3 x 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 :
--
\frac{90}{5} = 18
\frac{80}{5}  = 16
\frac{70}{5}  = 14
\frac{60}{5}  = 12
\frac{50}{5^{2}}  = 2
\frac{40}{5}  = 8
\frac{30}{5}  = 6
\frac{20}{5} = 4
\frac{10}{5}  = 2
--
So total highest power will be 9 and 1 add 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 x 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 x 5
Therefore the total number of five's are 7.
Hence the greatest positive power of 5 that divides 30! is 7.

166 x 6 + 5 = 1001 is the smallest 4-digit number gives a remainder of 5 when divided by 6.
So  N = 1001 + 6 x m   for some m.
so  N = 200 x 5 + 1 + 5 m + m
=  (200 + m) x 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 x 2 = 1013.

p=11k+7
so (p+4)(p+15)=(11k+11)(11K+22)
=11(k+1) x 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 x 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.