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Some similar divisibility properties are given. Moreover, we show that, for all positive integers m and n, the product $am{am+bm-1\\choose am}{an+bn\\choose an}$ is divisible by m+n. In fact, the latter result can be generalized to the q-binomial coefficients and q-integers case, which generalizes the positivity of q-Catalan numbers. We also propose severa"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1312.7548","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-12-29T15:18:41Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"f1748b41a7ec45f0d1d05ad0a402a9d84381a5ecb17fef0c46ff370f20e49f9c","abstract_canon_sha256":"b33f5127fbdd871bf32db5138a63ee502f2d1209598d0a544a3e997ee305fd0e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:03:31.352925Z","signature_b64":"23Q8kKmC86vyEqJ2W/e71JqP7/WKNA6SfmDc2hukLd2U/cKlRjO3KLaTHRgdqj6YwQHGMxTFyZV7XW9upEPhBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"089b546bd1bee78e87f73ddc40edd0c19b471561fa0c6903b770c549454a10a6","last_reissued_at":"2026-05-18T03:03:31.352475Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:03:31.352475Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Proof of two divisibility properties of binomial coefficients conjectured by Z.-W. 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