Question

What is the largest positive integer $n$ such that $1457$ and $1754$ leave the same remainder when divided by $n$

Answer 1

The largest positive integer n, that leaves the same remainder on dividing 1457 and 1754, is computed to be n = 297.

In the question, we are asked to find the largest positive integer n, such that 1457 and 1754 leave the same remainder when divided by n.

We assume the remainder to be r in both cases.

Now, 1457 - r is divisible by n.

We assume the quotient of (1457 - r)/n to be a.

Also, 1754 - r is divisible by n.

We assume the quotient of (1754 - r)/n to be b.

So we can write, 1457 = an + r,

or, an + r = 1457,

or, r = 1457 - an ... (i).

Similarly, 1754 = bn + r,

or, bn + r = 1754,

or, r = 1754 - bn ...(i).

From (i) and (ii), we can write that:

1754 - bn = 1457 - an,

or, 1754 - 1457 = bn - an,

or, 297 = n(b - a).

Now, for n to be largest, (b - a), needs to be minimum.

Thus, we take b - a = 1, to get n = 297.

Thus, the largest positive integer n, that leaves the same remainder on dividing 1457 and 1754, is computed to be n = 297.

Learn more about integers and remainders at

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