Moses' death "on one foot":
Moses was the leader of the Jewish people in the Biblical books of Exodus, Leviticus, Numbers, and Deuteronomy. While the Torah tells us that he died (Deuteronomy 34), it does not tell us when this happened. However, there are enough clues in the Bible to arrive at an answer of when his "yahrtzeit" (the anniversary of his death) is.
When did Moses Die?
Context: This is from the Biblical Book of Joshua, describing the crossing of the Jordan River into the Land of Israel. In the Bible, the first month is Nisan, because that's when we left Egypt. Therefore, the Israelites crossed the Jordan River on Nisan 10.
Context: This is from the beginning of the Biblical Book of Joshua, after Joshua receives Divine marching orders (literally). Here we see that the Israelites had 3 days to prepare before crossing the Jordan River. If they crossed on Nisan 10, then they received this order on Nisan 7.
Context: This is from the Biblical Book of Deuteronomy, right after Moses died. The Israelites mourned for Moses for 30 days on the steppes of Moab. We can assume that they did not make preparations to cross the Jordan River until after these 30 days ended. Therefore, if they began their preparations for crossing on Nisan 7, Moses must have died on Adar 7.
When is Moses' Yahrtzeit in a Leap Year?
The Torah says that Passover is to be in the spring (Exodus 23:15). Because the Jewish calendar is based on the moon, and 12 lunar months are 11 days shorter than 1 solar year, the Jewish dates would move earlier and earlier until they were no longer tied to their original season (as happens with the Muslim calendar). To keep Passover in the spring, the rabbis added an extra month every so often when the crops seemed to not be ready yet. This was put right before the end of the Biblical year, so instead of ending with Adar we had Adar 1 and Adar 2. The earliest mentions we have of this are in the Mishnah (Sanhedrin 1:2, explained on Sanhedrin 11a). Eventually, Hillel II fixed the calendar in 369 CE, establishing through mathematical calculations that there should be an extra month 7 out of every 19 years, in years 3, 6, 8, 11, 14, 17, and 19.
רַבִּי אוֹמֵר: אִם נִתְעַבְּרָה הַשָּׁנָה — יוֹצְאִין אַף עַל אֲדָר הַשֵּׁנִי מִפְּנֵי הַפּוּרִים.
Rabbi Yehuda HaNasi says: If the year was a leap year, the messengers go out also in the second Adar, due to Purim.
Context: This is from the Babylonian Talmud, Masechet (Tractate) Rosh Hashanah, which is about Rosh Hashanah (logically). Before the calendar was fixed, messengers had to be sent out before certain holidays to make sure everybody knew that the court in Jerusalem had decided it was a new month (based on witnesses of the moon). The question here is whether or not messengers had to go out before Adar 2 during a leap year to make sure people knew when to celebrate Purim. This is where we get the principle that anything that happens in Adar during a non-leap year is celebrated in Adar 2 during a leap year.
לָא, דְּכוּלֵּי עָלְמָא מִצְוֹת הַנּוֹהֲגוֹת בַּשֵּׁנִי — אֵין נוֹהֲגוֹת בָּרִאשׁוֹן
The Gemara rejects this argument: No, everyone agrees that the mitzvot observed in the second Adar are not observed on the first
Context: Same text. Here, the Gemara is saying that anything which happens in Adar 2 doesn't also happen in Adar 1.
Based on these texts, if Moses died on Adar 7, that would be observed during the second Adar in a leap year. Could he have died during the first Adar? No, because the math wouldn't work out. The people would have had to mourn for him for 60 days instead of 30.
When was Moses Born?
(1) Moses went and spoke these things to all Israel. (2) He said to them: I am one hundred and twenty years old now [literally "today"], I can no longer be active. Moreover, יהוה has said to me, “You shall not go across yonder Jordan.”
Context: This is from the Biblical Book of Deuteronomy, 3 chapters before Moses dies.
The Torah does not name the date when Moses was born. However, we can guess based on 2 assumptions:
1. Moses is saying the text cited above on the same day that he dies.
2. Moses is saying the text cited above on his birthday, rather than merely citing his age at some point during that year of his life.
Neither of these assumptions have to be true, especially the second one. However, if they are, then based on the previous calculations that put the date of Moses' death at Adar 7, then he would also be born on Adar 7.
During a leap year, Moses' birthday would be celebrated in the second Adar.
Appendix: Other sources on the matter
With appreciation to Brian Tice who compiled these in his source sheet "Zayin Adar"
Rabbi Aḥa bar Ḥanina says: That day was the sixth day of the month of Sivan. The ministering angels said before the Holy One, Blessed be He: Master of the Universe, should the one who in the future will receive the Torah on Mount Sinai on this day be stricken on this day? As this was also the date on which the Torah was received. The Gemara asks: Granted, according to the one who says that Moses was placed in the water on the sixth of Sivan, you find that there can be three months during which Moses was hidden after his birth; as the Master said (Tosefta 11:7): Moses died on the seventh of Adar, and Moses was born on the seventh of Adar. And based on this, from the seventh of Adar until the sixth of Sivan there are three months, which correspond to the three months Moses was hidden before being placed in the water. But according to the one who says that it was on the twenty-first of Nisan, how can you find that he was hidden for three months? The Gemara answers: That year was a leap year in which there were two months of Adar. Moses was hidden most of the first month of the three, from the seventh day of the first Adar when he was born, and most of the last month of the three, i.e., all of Nisan until the twenty-first, and the entire middle one. All of this together is considered as three months.
(ג) וילך משה וידבר את הדברים האלה אל כל ישראל. בן מאה ועשרים שנה אנכי היום. יתכן לומר כי היו ימיו ק"כ שנה כנגד ק"כ יום שעמד בהר, ארבעים יום ראשונים לקבל הלוחות, ארבעים יום שניים להתפלל על עון העגל, ארבעים שלשיים לקבל לוחות שניות, וזה טעם אנכי ולא אמר אני. ודרשו רז"ל היום מלאו ימי ושנותי, היום נולדתי והיום אמות. וכן אמר בתחלת דבריו (דברים כ״ט:ט׳) אתם נצבים היום, כל הענין דבק, והכל היה ביום מיתתו. והיום זה ז' באדר היה והוא סוף שנת הארבעים, בו ביום מת משה בו ביום נולד. בו ביום מת משה, הוא שכתוב בסוף התורה (שם לד) ויבכו בני ישראל את משה בערבות מואב שלשים יום ויתמו ימי בכי אבל משה, וראוי להדביק אל הכתוב הזה שבסוף התורה מה שכתוב בתחלת יהושע (יהושע א׳:י״א) ויצו יהושע את שוטרי העם הכינו לכם צדה כי בעוד שלשת ימים אתם עוברים את הירדן. ומי שמחבר הכתובים זה עם זה ימצא הענין מבואר כי לא צוה יהושע להכין צדה עד שתמו ימי בכי אבל משה, ולכך הוצרך הכתוב לומר ויתמו ימי בכי אבל משה, והזכיר שם הכתוב (שם ד) והעם עלו מן הירדן בעשור לחדש הראשון, ואם תמנה למפרע מעשרה בניסן ל"ג יום והם שלשה ימים לצדה ושלשים לאבלו של משה, תמצא יום מיתתו בז' באדר. בו ביום נולד משה, שכתוב (שמות ב׳:ג׳) ולא יכלה עוד הצפינו, והיה זה בששה בסיון, יום שהיה עתיד לקבל התורה שנתוסף בו עוצם אורה באותו יום לרמוז על גדולתו באותו יום, ועל כן לא יכלה עוד הצפינו, ועשתה תחבולה לשומו בתיבת גומא כדי להסתיר את גופו מפני תוקף אורו. ואם תחשוב מששה בסיון למפרע שלשה ירחים תמצא ים לידתו בז' באדר, ולכך אמר ירחים שהם חדשי הלבנה, והן אדר ניסן אייר, שנים חסרים ואחד מלא, ולא אמר חדשים לפי שהחדש חלק אחד משנים עשר שבשנה ואינן מוסיפין ולא גורעין, מה שאין כן חדשי הלבנה שהן מוסיפים וגורעין, ומפני זה הזכיר לשון ירחים כדי שיתבאר לנו מזה יום לידת הנביא. והנה זה מבואר מן הכתובים כי בז' באדר נולד משה ובו ביום מת וזהו מה שדרשו רז"ל שהקב"ה ממלא שנותיהם של צדיקים מיום ליום שנאמר (שם כג) את מספר ימיך אמלא.
(3) וילך משה וידבר את הדברים האלה אל כל ישראל. בן מאה ועשרים שנה אנכי היום, וגו', “Moses went and spoke these words to all of Israel. ‘I am 120 years old this day.’” It is possible that the 120 years Moses lived corresponded to the 120 days he spent on Mount Sinai. During the first 40 days he received the first set of Tablets. During the second 40 days he pleaded for G’d to forgive the people the sin of the golden calf; he spent the next 40 days on the mountain in order to receive the second set of Tablets. This may be the reason why he used the term אנכי instead of אני.
Our sages in Sotah 13 feel that he meant to say that on that day he was precisely 120 years old, i.e. that he died on the same day of the same month as he had been born. This may also be the reason why he used the word היום at the beginning of the previous Parshah, so that all these speeches took place on the same day. The day in question was the seventh day of Adar at the end of the Israelites 40th year in the desert. We have further proof of this in 34,8 where the Torah reported that the people mourned Moses for 30 days during which they did not journey. The Torah states explicitly that the days of Moses’ mourning came to an end. It is proper to read this verse in conjunction with Joshua 1,10: “Joshua commanded the law-enforcement officers of the people: ‘prepare for yourselves provisions for in another three days you will cross the river Jordan.’” Anyone reading these two verses together will realize that Joshua would not command the people to prepare these provisions until the mourning period for Moses had passed. To make this plain, the Torah had to write the verse: “the days of crying for (the death of) Moses had concluded.” The Book of Joshua 4,19 reports that the Jewish people completed crossing the river Jordan on the tenth of the first month, i.e. 3 days after the conclusion of the mourning period for Moses. By counting backwards you will arrive at the date of the 7th Adar as the date on which Moses had died. When the Torah in Exodus 2,3 reported that Moses’ mother was unable to hide him any longer, the date referred to is the 6th of Sivan, the day on which the revelation at Mount Sinai took place 80 years later. On that day, the light which filled only his parents’ home on the day he was born became so powerful that it shone for the whole Jewish nation. The light which had filled the house of Amram had already forecast the eventual greatness of the boy that had been born to Yocheved. The reason Yocheved had to hide Moses in a basket, well insulated, was that the light surrounding the baby had become so much stronger that it represented a danger to the family. The Torah speaks of Yocheved having hidden Moses for שלשה ירחים ‘three lunar months,” rather than שלשה חדשים, “three months,” (Exodus 2,2) to indicate that these were lunar months. The months Adar, Nisan, and Iyar are 29,30, and 29 days long, respectively. This gives us a total of 88 days during which Moses was kept “hidden” at home, or it takes us back to the 7th day of Adar as Moses’ birthday. From all this it becomes clear that Moses died on the same day of the same month on which he had been born. The word חדש implies a period which equals 1/12th of the year, something not subject to variation either by adding or subtracting from it. The same is not true of the period known as ירח, a period which may comprise either 29 or 30 days. The reason why the Torah used the term ירחים in the verse mentioned was to enable us to calculate the precise date on which Moses was born. This prompted our sages in Kiddushin 38 to state that G’d completes the life cycle of the righteous from day to day (to grant them whole years) something they base on Exodus 23,26: את מספר ימיך אמלא, “I will fill (round out) the number of your days (meaning “years”).