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"je les ai achetés" is an exemple of the one exception to the rule that you mentioned: that the past participles of "avoir" verbs are invariable- i.e they never add endings to agree with the subject, unlike "être" verbs, whose past participles add endings "e", "s" or "es" to agree with the gender and number of their subject.
e.g
j'ai mangé je suis alléeil a mangé il est alléelle a mangé elle est alléenous avons mangé nous sommes alléselles ont mangé elles sont allées so why does "je les ai achetés" make grammatical sense?
-it is something called the preceding direct object pronoun-you are probably already familiar with "direct object pronouns"- if not they are simply the pronouns that are used in place of a noun in the accusative (the direct object of the sentence)
e.g
· "je mange le pomme" = I eat the apple "je le mange" = I eat it LEis the direct object pronoun (MASCULINE SINGULAR)
· "je vois la femme" = I see the woman "je la vois" = I see her LA is the direct object pronoun (FEMININE SINGULAR)
· "j'écoute les oiseaux" "je les écoute" LESis the direct object pronoun (MASCULINE PLURAL) (the DOP for feminine . plural is also "les)
IN THE PAST TENSE, WHEN THE DIRECT OBJECT PRONOUN PRECEDES THE VERB, THE PAST PARTICIPLE MUST AGREE
e.g · j'ai vu la femme --- jel'ai vue(+ "e" because "la femme" or "la" are feminine singular) · j'écoute les oiseaux --- jelesai écoutés(+ "s" because "les oiseaux" or "les" are masculine plural) · je mange le pomme --- je l'ai mangé (no ending because "le pomme" or "le" are masculine singular) · j'ai porté les jupes --- je lesai portées(+ "es" because "les jupes" or "les" are feminine plural"
therefore:-· "je les ai achetés" - the "les" must be referring to a masculine plural direct object, represented by the direct object pronoun "les". – · The "les" could represent any masculine plural noun, e.g. "les vêtements" or "les pommes".· Because the direct object pronoun "les" preceeds the verb "ai acheté", the past participle "acheté" must agree to the masculine plural "les" by adding an "s".