Solution:
(a) The difference between two integers will be 1, only if one is even and the other one is odd. 4x will always be even, so 17 y has to be odd and hence y has to be odd.
Moreover, the number 17 y should be such a number that is 1 less than a multiple of 4. In other words, we have to find all such multiples of 17, which are 1 less than a multiple of 4. The first such multiple is 51. Now you will find that as the multiples of 17 goes on increasing, the difference between it and its closest higher multiple of 4 is in the following pattern, 0, 3, 2, 1, e.g., 52 − 51 = 1,
68 − 68 = 0, 88 − 85 = 3, 104 − 102 = 2, 120 − 119 = 1, 136 − 136 = 0
So the multiples of 17 that we are interested in are 3, 7, 11, 15.
Now Since, x ≤ 1000, 4x ≤ 4000. The multiple of 17 closest and less than 4000 is 3995 (17 × 235). And incidentally, 3996 is a multiple of 4, i.e. the difference is 4.
This means that in order to find the answer, we need to find the number of terms in the AP formed by 3, 7, 11, 15, ..., 235, where a = 3, d = 4.
Since, we know that T = a + (n − 1)d, so 235 = 3 + (n − 1) × 4.
Hence, n = 59.
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