Successive Division

Number Theory Level 2

A number is successively divided by 16, 12 and 10 leaving 2, 10 and 8 as remainders respectively. The sum of remainders when order of divisors be reversed is:

Clarification : After dividing the given number by 16, the quotient that I've got from this division is divided by 12 and, then, the quotient that I've got from this second division is divided by 10 to get the respective remainders.

a prime number a transcendental number a complex number a composite number

1 solution

Prakkash Manohar
Feb 26, 2014

Let the number be n.

n = 16a + 2 (a= quotient) -----------(1)

a = 12b + 10 (b=quotient) -----------(2)

b = 10c + 8 ------------(3)

From (1), (2) and (3), we get

n = 1920c + 1698

When order of divisors is changed,

n/10 = (1920c + 1698)/10

(192c + 169) = quotient and remainder = 8

Now, (192c + 169)/12

(16c + 14) = quotient and remainder = 1

Now, (16c + 14)/16

c = quotient and remainder = 14

So, sum of remainders = 8 + 1 + 14 = 23 which is a prime number.

Note: Transcendental numbers are those numbers which cannot be a solution to a polynomial equation with rational coefficients. eg. π (pi), e (Euler's number), Liouville Constant, etc.

as the residual values are even after dividing the even no so it must be composite no. and the number is N = 16x +2 x = 12y +10 y = 10z+8 Replacing N = 1920z + 1698 for z = 1 solution is N= 3618 for z = 2 solution is N= 5538 for z = 3 solution is N= 7458 for z = 4 solution is N= 9378 and so on ..

Moshiur Mission - 7 years, 3 months ago

16 12 10 2 10 8

So N = required number = ( (8x12) +10) x16 +2) = 1698 (least possible number)

Now divide 1698 successively by 10,12,16 , we get 8, 1, 14 as remainder . Now sum of remainder = 8+1+14 = 23 = prime number.

Pankaj Pandey - 4 years ago

Successive Division

1 solution

0 pending reports