Two friends were born in the 20th century, a week apart and in the same month and year. Writing from left to right the date in the form of the (or the) digit(s) of the day, (or the) digit(s) of the month, and the last two digits of the year, we get two numbers, one being six times the other. We do not put a 0 in front of the first nine months. What is the birth date of the older friend?
The two friends were born in the same month and year, with a difference of 7 days, so one was born on $d / m / a$ and the other on $(d+7) / m / a$. With these dates, we form the numbers $(d)(m)(a)$ and $(d+7)(m)(a)$. We know that:

$$
(d+7)(m)(a)=(d)(m)(a)+7 \times 10^{k}
$$

Thus,

$$
(d+7)(m)(a)=6 \times(d)(m)(a), \quad \Longrightarrow \quad 7 \times 10^{k}=5(d)(m)(a)
$$

Therefore, $k=3$ if the month has 1 digit and $k=4$ if the month has 2 digits. In the first case, when $k=3$, we have $\frac{7000}{5}=1400$, that is, April 1, 1900. Therefore, his friend was born on April 8, 1900. In the second case, when $k=1, \frac{70000}{5}=14000$ is not a valid date.