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# Solution

The crux of the question is to find the greatest number that will divide 34, 43, 70, 124, and 358 to leave the same remainder.

A one-step solution to the problem is that this greatest divisor will be the HCF of (43 - 34), (70 - 43), (124 - 70), (358 - 124), and (358 - 34); i.e., the HCF of 9, 27, 54, 234, and 324.

Now, 9 = 3 × 3

27 = 3 × 3 × 3

54 = 2 × 3 × 3 × 3

234 = 2 × 3 × 3 × 13.

324 = 2 × 2 ×3 × 3 × 3 × 3.

So, the HCF = 9.

The total number of seats in each row is equal to the HCF. Therefore, there are nine seats in each row. There are 20 rows. So, the seating capacity of the aeroplane = 9 × 20 = 180.

But the question is, why am I considering the HCF of the differences?

That calls for some explanation.

When a number N is divided by the greatest divisor "d", we get a quotient "q" and a remainder "r".

Then, N = dq + r. For example, when 21 is divided by 4, we get 5 as the quotient and 1 as the remainder. Then, 21 = 4 × 5 + 1.

So, N = dq + r. Let me call this Equation I.

If the remainder is 0, it implies N = dq. Let me label this as Equation II.

If "n" divides N1 and also N2, then it also divides (N1 + N2). As an example, 3 divides 6 and 3 divides 18, so 3 divides 18 + 6, i.e. 24. Let me label this as Statement III.

Similary, if "n" divides N1 and N2, "n" also divides (N1 - N2). Let me label this as Statement IV. For example, 3 divides 12 and 3 divides 18; so, 3 divides (18 - 12), i.e. 6.

Now, we get back to our original problem. Let "d" be the greatest divisor that divides 34, 43, 70, 124, and 358 to leave the same remainder, say "r". The quotients are, of course, different. Let me label the quotients as q1, q2, q3, q4, and q5.

Using Equation I, we get:

34 = dq1 + r

43 = dq2 + r

70 = dq3 + r

124 = dq4 + r

358 = dq5 + r

We may rewrite them as follows:

34 - r = dq1

43 - r = dq2

70 - r = dq3

124 - r = dq4

358 - r = dq5

Using Equation II, this implies (34 - r), (43 - r), (70 - r), (124 - r), and (358 - r) are completely divisible by "d", leaving no remainders.

We want to get rid of "r". Using Statement IV, we conclude "d" is the greatest divisor that divides [(43 - r) - (34 - r)], [(70 - r ) - (43 - r)], [(124 - r) - (70 - r)], [ (358 - r) - (124 - r)], and [(358 - r) - (34 - r)].

This implies "d" is the greatest divisor of (43 - r - 34 + r), (70 - r - 43 + r), (124 - r - 70 + r), (358 - r - 124 + r), and (358 - r - 34 + r), i.e., 9, 27, 54, 234, and 324. This is what we had set out to check.

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