# Task №3

What two digits can be appended to the number 1313 on the right so that the resulting six-digit number is divisible by 53?
Answer: 34 or 87.

Solution: $1313 x y$, where x is the tens digit, and y is the units digit, then

$10 \leq \mathrm{xy} \leq 99, \quad 1313 \mathrm{xy}=\underline{131300}+\mathrm{xy}=\underline{2477 \cdot 53+19}+\mathrm{xy}$. Since the number is natural and must be divisible by 53, then ( $19+$ xy) must be a multiple of 53, that is, the possible cases are:

1) $19+x y=53$, then $x y=34$
2) $19+x y=106$, then $x y=87$
3) $19+x y=159$, hence $x y=160-$ does not fit, as it contains not two, but three digits.