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We know from Gen 37 that Yosef was 17 when he went out to meet his brothers and ended up on the way to Egypt, and we know from Gen 41:46 that he was 30 when Paro appointed him as second in commend. We also know that this was two years after he, in prison, interpreted the dreams of the butler and baker.

So the earliest he could have gone to prison is 17 and the latest is 28. 17 seems unlikely, as he needed time to rise to power in Potiphar's house, but there's still a decade or so of uncertainty.

So how old was Yosef when Potiphar's wife accused him and he got sent to prison? Rashi is silent on the question and none of the various chumashim in use at our torah-study group had anything.

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The three oldest sources that I can find on this question are Shemot Rabbah 7:1, Pirqei deRebi Eliezer 39 and Seder Olam Rabbah 2. They read as follows:

הדא הוא דכתיב בכל עצב יהיה מותר ודבר שפתיים אך למחסור. בכל הדברים שאדם נושא ונותן בהם דברי תורה הוא נוטל עליהן שכר יכול אף בדברי בטלה כן תלמוד לומר ודבר שפתיים אך למחסור. אתה מוצא לא היה יוסף ראוי לנתן בבית האסורין אלא עשר שנים מפני שהוציא דבה על עשרת אחיו ועל ידי שאמר לשר המשקים כי אם זכרתני אתך והזכרתני אל פרעה נתוסף לו עוד שתי שנים

Here it is written, "From all toil there is some gain, but idle chatter is pure loss" (Proverbs 14:23, JPS). Should one's business matters concern words of Torah, he receives reward for them. It is likewise with words of idleness, as it says: "but idle chatter is pure loss". So you find that Joseph was only deserving of being placed in prison for ten years, since he delivered an evil report of his ten brothers, but due to the fact that he said to the chief cupbearer, "But think of me... mentioning me to Pharaoh" (Genesis 40:14, JPS), a further two years were added to his sentence.

  • Shemot Rabbah 7:1

והביאה עליו האשה עלילות דברים ונחבש בבית האסורים עשר שנים

And the woman brought false charges against him and he was locked up in the prison for ten years.

[Rabbi David Luria (19th c. Lithuanian), in his peirush on PRE, suggests that this is a reference to the length of time that Joseph spent in prison before the extra two years were added.]

  • Pirqei deRebi Eliezer 39

ירד יוסף למצרים ועשה בבית פוטיפרע י"ב חדשים שנא' ויהי מאז הפקיד אותו וגו' בבית מפני החמה ובשדה מפני הצינה ועשה בבית האסורים י"ב שנה שנא' ענו בכבל רגליו וגו' עד עת בא דברו וגומר

Joseph went down to Egypt and worked in Potiphar's house for twelve months, as it says: "And from the time that [the Egyptian] put him in charge... [the blessing of the Lord was upon everything that he owned, in the house and outside]" (Genesis 39:5, JPS). In the house because of the sun [ie: the summer]; and outside because of the cold [ie: the winter]. And he worked in the prison for twelve years, as it says: "His feet were subjected to fetters... until his prediction came true..."(Psalm 105:18-19, JPS).

[The word for "his feet", רגליו, is so spelt in the text but is to be read as "his foot", רגלו. According to Rabbi David Luria (Pirqei deRebi Eliezer 39:29), the midrash is deriving ten years from the value of the extra yodh in the ketiv. For a similar drash, see the Baal haTurim, Genesis 39:20, who derives Joseph's ten years in prison - before the addition of a subsequent two - from the extra yodh in the ketiv there.)

  • Seder Olam Rabbah 2

So, to answer your question, at the time of Joseph's arrest he was:

Seder Olam Rabbah: 18 years old;

Shemot Rabbah: 18 years old;

Pirqei deRebi Eliezer: 20 years old (acc. to R' David Luria, 18 years old).

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The Sedor Hadoros (2217) brings several opinions: The Sefer Hayashar (Parshas Vayeshev) and Seder Olam both maintain he was only in Potifar's house for a year, and spent the next twelve years on prison. The Yefeh Toar (on Midrash Rabba Parshas Behaloshcha and Vayera) questions this, and the Tzemach Dovid suggests he worked for Potifar for 11 years, and spent just two years in prison.

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Where is the Tzemach Dovid? –  Gershon Gold Feb 10 '13 at 2:02
    
How do they know? Tradition? –  Double AA Feb 10 '13 at 2:08
    
@DoubleAA Tzemach Dovid suggests that the "end of two years" (Parshas Miketz) could be (not from interpreting the dreams but) from when was put in jail. –  Michoel Feb 10 '13 at 2:16
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Baal HaTurim says he was in prison a total of 12 years

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How does he know? Tradition? –  Double AA Feb 10 '13 at 2:08
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