Which among the following can be expressed as (10p + q) (10q + p), where p and…

For any integers p and q — not necessarily single digits — let a = 10p + q and b = 10q + p; b swaps the coefficients of p and q, the same pattern as reversing a two-digit number's digits when p, q happen to be 0-9. The product expands algebraically to (10p + q)(10q + p) = 10p2 + 101pq + 10q2. Since a + b = 11(p + q) and a − b = 9(p − q), a target number N is expressible in this form ONLY IF N has some factor pair (a, b) — checked across ALL its divisor pairs, not only ones that look like two-digit reversals — whose sum is a multiple of 11 AND whose difference is a multiple of 9; p and q are then recovered as p = (10a − b) / 99 and q = (10b − a) / 99.

1. Check 1207: its factor pair 17 × 71 gives sum 17 + 71 = 88 = 11 × 8 (a multiple of 11) and difference 71 − 17 = 54 = 9 × 6 (a multiple of 9) — both required conditions hold.

2. Recover p and q from a = 17, b = 71: p = (10a − b) / 99 = (170 − 71) / 99 = 1 and q = (10b − a) / 99 = (710 − 17) / 99 = 7, both integers.

3. So p = 1, q = 7 gives 10p + q = 17 and 10q + p = 71, and 17 × 71 = 1207 exactly — the required form is satisfied.

4. Checking every divisor pair of the other three values the same way: for 1344, three pairs have a sum divisible by 11 — (3, 448) with sum 451, (8, 168) with sum 176, and (14, 96) with sum 110 — but none of their differences (445, 160, and 82) is divisible by 9; for 1728 and 1296, no divisor pair at all has a sum divisible by 11. So none of 1344, 1728, or 1296 has any factor pair meeting both conditions, for any integers p and q — not only when p, q are restricted to single digits.

Cross-check by substituting p = 1 and q = 7 directly into the expansion: 10p2 + 101pq + 10q2 = 10(1) + 101(7) + 10(49) = 10 + 707 + 490 = 1207, matching independently.

So 1207 is the only one of the four values expressible as (10p + q)(10q + p) for integers p and q.