Question

What is the largest number that divides 245 and 1029, leaving remainder 5 in each case?
(a) 15
(b) 16
(c) 9
(d) 5

Answer 1

(b) 16
Clearly the required number divides (245 - 5), i.e. 240 and (1029 - 5), i.e. 1024 exactly.
So, the required number is H.C.F. (240, 1024).
Now,
240 = 2 × 2 × 2 × 2 × 3 × 5 = 2⁴ × 3 × 5
1024 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2¹⁰
Therefore, H.C.F. of 240 and 1024 = 2 × 2 × 2 × 2 = 16
Hence, 16 is the largest number that divides 245 and 1029 and leaves remainder 5 in each case.