141 Let $a, b$ be positive integers, proposition “甲” represents “the number $a^{2}+b^{2}$ is divisible by 8”, proposition “乙” represents “the number $a^{3}+b^{3}$ is divisible by 16”, then which of the following four judgments is correct?
A. 甲 is a sufficient condition for 乙, but not a necessary condition
B. 甲 is a necessary condition for 乙, but not a sufficient condition
C. 乙 is a sufficient and necessary condition for 甲
D. 乙 is neither a sufficient condition nor a necessary condition for 甲
141 A. The square of an odd number leaves a remainder of 1 when divided by 8, so when at least one of \(a\) or \(b\) is odd,
$$
a^{2}+b^{2} \equiv 2 \text { or } 1 \text { or } 5(\bmod 8) .
$$

Thus, if condition 甲 holds, \(a\) and \(b\) are both even, and we can set
$$
a=2 k, b=2 h
$$

Then \(a^{2}+b^{2}=4\left(k^{2}+h^{2}\right) \equiv 0(\bmod 8)\), so \(k\) and \(h\) have the same parity. Therefore,
$$
a^{3}+b^{3}=8\left(k^{3}+h^{3}\right) \equiv 0(\bmod 16)
$$

Thus, 甲 is a sufficient condition for 乙.
If we take \(a=2^{4}-1, b=2^{4}+1\), then \(a^{3}+b^{3} \equiv(-1)^{3}+(1)^{3} \equiv 0(\bmod 16)\). However, \(a^{2}+b^{2}=\left(2^{4}-1\right)^{2}+\left(2^{4}+1\right)^{2}=2^{9}+2\) is not divisible by 8, so “乙” is not a sufficient condition for “甲”, and “甲” is not a necessary condition for “乙”.