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Question Number 211454 by BaliramKumar last updated on 09/Sep/24

	Answered by A5T last updated on 09/Sep/24

	$L e t N = p_{1}^{2} \times p_{2}^{4}; t h e n i t h a s (2 + 1) (4 + 1) = 15 d i v i s o r s$ $\Rightarrow N^{2} = p_{1}^{4} \times p_{2}^{8}; t h e n i t h a s (4 + 1) (8 + 1) = 45$
	Answered by mr W last updated on 11/Sep/24

	$c a s e 1 : N = p^{a}$ $a + 1 = 15 \Rightarrow a = 14$ $p ⩾ 2 \Rightarrow N ⩾ 2^{14} = 16384 > 9999$ $\Rightarrow n o s o l u t i o n$ $c a s e 2 : N = p^{a} q^{b}$ $(a + 1) (b + 1) = 15 = 3 \times 5 \Rightarrow a, b = 2, 4$ $2^{2} 5^{4} = 2500 ✓$ $2^{2} 7^{4} = 9604 ✓$ $3^{2} 5^{4} = 5625 ✓ 5^{2} 3^{4} = 2025 ✓$ $3^{4} 7^{2} = 3969 ✓$ $\Rightarrow t h e r e a r e 5 s o l u t i o n s$ $N^{2} = p^{2 a} q^{2 b} h a s (2 a + 1) (2 b + 1)$ $= 5 \times 9 = 45 d i v i s o r s$
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