Question

how many natural nui divide 2500,4250,6700 leaving the same remainder in each case

Answer 1

Let N is the largest number, if we divide 2500 , 4250 and 6700 by N leaving the same remainder r in each cases.
from Euclid lemma ,
2500 = N × g + r ...........(1)
4250 = N × f + r ............(2)
6700 = N × h + r...........(3)

from equations (1) and (2),
4250 - 2500 = N(f - g)
1750 = N(f - g)
350 × 5 = N(f - g) .........(4)
similarly, from equations (2) and (3),
6700 - 4250 = N(h - f)
2450 = N(h - f)
350 × 7 = N(h - f) ........(5)

and from equations (3) and (1),
6700 - 2500 = N(h - g)
4200 = N(h - g)
350 × 12 = N(h - g)........(6)

from equations (4), (5) and (6),
N = 350
now we have to find prime factor of N
prime factor of 350 = 1 × 2 × 5 × 5 × 7
= 1 × 2 × 5² × 7

so, number of ways to arrange {1, 2, 5² , 7} = 4!/2!

hence, total number of solutions = 12