#May25 #MathsbyAmiya

a = 13x, b = 13y

(x, y) = co prime

a³ = 13³x

Let x is not a multiple of 13

if y is not a multiple of 13

GCD(a³, b) = 13

If y is multiple of 13

GCD(a³, b) = 13²

If y is a multiple of 13²

GCD(a³, b) = 13³

Sum = 13 + 169 + 2197

= 2379

A, B and C went to a game shop to play a game. They took total 8 coins. These coins were
to be distributed among A, B and C. In how many ways can these coins be distributed so that
A will get at least 2 coins, B will get at least 1 coin and C will get at least 3 coin

Unit digit prime = 2,3,5,7
2,5 not possible so 3 , 7
X3 , X7
X3 = 13 , 23 , 43 , 53 , 73 , 83
X7 = 17 , 37 , 47 , 67 , 97

Total = 6 + 5 = 11

Ye scale kya h yha?

Scale means base system..
Let number be ABC in base system 7.
(ABC)₇ = (CBA)₉
so a,b,c < 7 ..

49a+7b+c = 81c+9b+a
48a = 2b + 80c
24a-40c = b
8(3a-5c) = b
B can be multiple of 8 only so either 0,8.
since the base system is 7 so B = 8 is not possible.
Therefore only B = 0 is possible..

8(3a-5c) = 0
3a=5c
a : c = 5 : 3
so a = 5 , c = 3 same logic a,c <7

(503)₇ = 7²*5 + 7¹*0 + 7⁰*3
= 245 + 3 = 248 ✅

333....(36 times) == 0
333*10³³ + k = 0
k = -10*10³³ = -10³⁴ = -10¹⁶
k = -(10⁴) = -6⁴ = -(36*36) = -4
k = 15

Let's take 5 times 3
33333 to make it 33

Subtract 30000-3000-300
=33300= (333) *10^(n-3)


Same make it (3333... 15 times)
18 time divisible by 19
Subtract -333*10^15
(-10^16/19) =-5^8/19
=-216*6/19=-(42/19) =15 reminder

23.

123456 123456.....(7times)
Is divisible by 7

Because 7*(456-123)


Reminder (123456123456) /7

=1