What is the largest number that divides 245 and 1029, leaving remainder 5 in each case?
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

Solution:

(b) 16

We know that the required number divides 240 (245 − 5) and 1024 (1029 − 5).
∴ Required number = HCF (240, 1024)

$240=2 \times 2 \times 2 \times 2 \times 3 \times 5$

$1024=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$

$\therefore \mathrm{HCF}=2 \times 2 \times 2 \times 2=16$

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