Theory of Numbers Test - 2

Any odd power of 9 has 9 at the unit place.
Thus, the remainder, when 927 is divided by 5, is 4.

Let x be the number.
Given: x = 296n + 75 [where n is the quotient]
Or, x = 37 × 8n + 37 × 2 + 1
Or, x = 37(8n + 2) + 1
Hence, when x is divided by 37, it will give 1 as the remainder.

412 ÷ 57
⇒ (43)4 = (57 + 7)4
Dividing by 57, we get remainder 74; but 74 can't be the remainder as it is more than 57.
Now, apply the same theorem to 74.
73 = 343
73 ÷ 57, Remainder = 1
Remainder when 73 × 7 is divided by 57,
1 × 7 = 7
So, the remainder is 7.

Sum of all the numerals of given number = 2 (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) = 2 (45) = 90, i.e. this number is divisible by 9.
Thus, we can write (1234567890123456789)24 as (9K)24. Similarly, 6561 can be written as 94.
This means that the given number will be completely divisible by 6561, and remainder will be
0.

If x and y are co-primes, they don't have any common factor. Hence, their squares are also co- primes.

721 + 722 + 723 + 724 is divided by 25, i.e.

721(70 + 71 + 72 + 73)

= 721(1 + 7 + 49 + 343)

= 721(400)

i.e. when 721(400) is divided by 25, remainder = 0 (as 400 is divisible by 25)

Co-primes are the numbers that do not have any common factor other than 1.
So, the numbers must be 1 and 8.
∴ Sum = 1 + 8 = 9

Co-Prime of 21 less than 21 is 1,2,4,5,8,10,11,13,16,17,19 and 20.