The number of common terms in the two sequences 17, 21, 25,…, 417 and 16, 21, 26,…, 466 is
The terms of the first sequence are of the form 4p + 13
The terms of the second sequence are of the form 5q + 11
If a term is common to both the sequences, it is of the form 4p+13 and 5q+11
or 4p = 5q -2. LHS = 4p is always even, so, q is also even.
or 2p = 5r - 1 where q = 2r.
Notice that LHS is again even, hence r should be odd. Let r = 2m+1 for some m.
Hence, p = 5m + 2.
So, the number = 4p+13 = 20m + 21.
Hence, all numbers of the form 20m + 21 will be the common terms. i.e 21,41,61,...,401 = 20.
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