Which of the following numbers will replace the question marks (?) in the given series? 61, ?, ?, 73, 79, 83, 89

Correct — D. Read the terms you are given and ignore the gaps for a moment: 61, 73, 79, 83, 89.

Rule: consecutive prime numbers.

There is no constant difference to find — 73 to 79 is +6, 79 to 83 is +4, 83 to 89 is +6. What every visible term has in common is that it is prime.

After 61 the next two primes are 67 and 71, and 73 follows them, which is exactly what the series shows: 61, 67, 71, 73, 79, 83, 89 → option (d).

Why the others are wrong

• (a)
  67, 69 — 67 is prime but 69 = 3 x 23, so it is composite. One composite term is enough to break a run of primes.
• (b)
  63, 65 — Both are composite: 63 = 7 x 9 and 65 = 5 x 13. Neither can stand between 61 and 73 in a list of primes.
• (c)
  63, 67 — This pair also carries 63 = 7 x 9, and it skips 71 — the prime that has to sit between 67 and 73.

A prime has exactly two factors, 1 and itself. Series built on primality look ordinary, because the gaps between primes are irregular and there is no step to spot.

That irregularity is itself the signal. When the differences refuse to settle into a pattern, stop differencing and ask what property the terms share.

The primes from 60 to 90 are 61, 67, 71, 73, 79, 83 and 89. Knowing that short list outright turns this question into a five-second read.

Key facts

• A prime number has exactly two factors: 1 and the number itself.
• The primes between 60 and 90 are 61, 67, 71, 73, 79, 83 and 89.
• 63 = 7 x 9, 65 = 5 x 13 and 69 = 3 x 23, so none of those three is prime.
• Gaps between consecutive primes are irregular, so a prime series has no constant difference.

Common traps

• Hunting for a constant or growing difference and filling +4, +8 style gaps
• Accepting 69 as prime because it is odd and does not look familiar

No directly related past PYQ was found.