if 126 power of 7 is divided by 48 find remainder?

Question

explain 2 ki power 2=4 in the same way 7ki power 126 is here

Answer 1

Hmm... are you trying to find the remainder of (7^126)/48? Everyone else seems to be interpreting your question a different way.

If that's what you mean, then:

First, note that 48 = 7^2 - 1

7^2 - 1 is a factor of 7^(2n) - 1 for any positive integer n.

Therefore 7^2 -1 is a factor of 7^126 - 1. So, the remainder when 48 is divided into 7^126 has to be 1.

* * * * * *

Uh, for all the folks solving this as 126^7: it's a pretty trivial question in this case, isn't it?

Also, to HIMANSHU (a few posts down): First, not every coefficient of the binomial expansion is going to be divisible by 48. Secondly, 48 - 1 = 49 ??

Answer 2

mbdwy's answer is correct. mitch's answer is wrong. Remainder = 32. Have you learn modulo yet? Let a, b, n be integers. a ≡ b (mod n) means that a - b = n*k where k is an integer. In english, it means that b is the remainder when a is divided by n. You want to find b such that 128^7 ≡ b (mod 48) There are 3 theorems regarding modulo. They are: If a ≡ b (mod n), then a^2 ≡ b^2 (mod n) If a ≡ b (mod n) and c ≡ d (mod n), then ac ≡ bd (mod n) If a ≡ b (mod n) and b ≡ c (mod n), then a ≡ c (mod n) Let us take a look at 128 ≡ 32 (mod 48). It is easy to calculate 32 in this case. Using the first theorem, you get 128^2 ≡ 32^2 (mod 48) We can calculate that 32^2 ≡ 16 (mod 48). Therefore, using the third theorem, we get 128^2 ≡ 16 (mod 48) Using the first theorem, we get 128^4 ≡ 16^2 (mod 48) Using the third theorem, 16^2 ≡ 16 (mod 48) 128^4 ≡ 16 (mod 48) So, we have 128^2 ≡ 16 (mod 48) 128^4 ≡ 16 (mod 48) Using the second theorem this time, we get 128^6 ≡ 16^2 (mod 48) 128^6 ≡ 16 (mod 48) We now have 128 ≡ 32 (mod 48) 128^6 ≡ 16 (mod 48) Using the second theorem again, we get 128^7 ≡ 32*16 (mod 48) We can calculate that 32*16 ≡ 32 (mod n). Therefore using the third theorem we get 128^7 ≡ 32 (mod 48) Hence your remainder is 32.

Answer 3

126 Divided By 7

Answer 4

Think about it this way: if 48 divides into 126^7 evenly, then the remainder will be zero. You don't have to do the big multiplication of 126 x 126 x 126 x 126 x 126 x 126 x 126. All you have to do is look at the factors involved. If 48's factors are in 126^7, then you're home free. 48 is 16 x 3, which is 2^4 x 3. So as long as you have at least four 2s and one 3, they divide evenly and you're OK. As it turns out, 126 factors into 2 x 3^2 x 7. Raising that to a power of 7 gives you 2^7 x 3^14 x 7^7. In other words, you're multiplying a string of seven 2s, fourteen 3s, and seven 7s. Since all we need are four 2s and one 3, we're good to go and everything works out with no remainder.

If 126^7 is to big to get your mind around, you can start at 48 and work your way bigger. 48 x 7 is 336, so 48 divides evenly into 336 and the remainder is 0. Factored out, that's 2^4 x 3 x 7.

48 x 42 is 2016. That means that if you divide 2016 by 48, you'll get 42, with a remainder of 0. If you just saw 2016 by itself, you wouldn't necessarily know that, but if you factored it out, you'd see that 2016 = 2^5 x 3^2 x 7. Any number whose factors have at least what's in 48 (i.e., four 2s and one 3), will be divisible evenly by 48. 126^7 is the same thing, just on a bigger scale.

Answer 5

126^7/48
=(63*2)^7 / (16*3)
=[3^2*7*2]^7 / 2^4*3 (when raising a power to a power you mult.)
=3^14 * 2^7 * 7^7 / 3 * 2^4 (when dividing exponents you subtract)
=3^13 * 2^3 * 7^7
=equals a whole number, remainder being 0

Answer 6

The problem can be reduced as follows:

126=48*2+30

So all we need to know is the remainder when 30 is raised to the 7th power.

48 divides 30^7 so the remainder is 0.

Answer 7

We can solve the problem as follows:
7^126 =(7^2 )^63
= 49^63
=(48 + 1)^63
Using Binomial Expansion ,we have:
(48+1)^63 = 48^63 *63C0 + 48^62 * 63C1 + 48^61 * 63C2 + .................... +48^1*63C62 +48^0 *63C63
or (48+1)^63 = 48^63 + 48^62 * 63C1 + 48^61 * 63C2 + .................... +48*63 + 1.
Since Each term is divisible by 48 ,Except last term ,i.e., 1,
therefore remainder= 1
Therefore ,remainder = 1.

Also, to Zanti3 , a few posts above :
Sorry for the second mistake but the first one you pointed out isn't a mistake at all.
I have written each term is divisible by 48 and not each binomial coeffecient ( nCr) and I think You are smart enough to understand that each term accept last term ,i.e., 1 is divisible by 48 because along with Binomial Coeffecients (which are natural numbers) there are also certain powers of 48 greater than or equal to 1 which make each term except 1 divisible by 48.

Hope You got it!

Answer 8

126 = 2 * 3 * 3 * 7
48 = 2 * 2 * 2 * 2 * 3

126 ^ 7 will have seven 2's, and fourteen 3's, as 48 has 4 2's and one 3 in it's factorization, it divides 126 ^ 7, moreover it divides 126 ^ 4

Answer 9

126^7(mod 48) = 0

Answer 10

Remainder 2