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Erd\\H{o}s asked whether $A(x)/x=\\exp(-(c+o(1))\\sqrt{\\log x}\\log\\log x)$ for some constant $c>0$. We prove that this holds with $c=1/(2\\sqrt{\\log 2})$; equivalently, $\\log(x/A(x))/(\\sqrt{\\log x}\\log\\log x)$ tends to $1/(2\\sqrt{\\log 2})$. The lower bound is obtained from primes in disjoint logarithmic intervals using a fourth-moment argument based on the multiplicative large sieve and a subset-product second moment. 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Let $A(x)$ count the positive integers $n\\le x$ such that, for every prime $p\\mid n$, there is a divisor $d>1$ of $n$ with $d\\equiv 1 \\pmod p$. Erd\\H{o}s asked whether $A(x)/x=\\exp(-(c+o(1))\\sqrt{\\log x}\\log\\log x)$ for some constant $c>0$. We prove that this holds with $c=1/(2\\sqrt{\\log 2})$; equivalently, $\\log(x/A(x))/(\\sqrt{\\log x}\\log\\log x)$ tends to $1/(2\\sqrt{\\log 2})$. The lower bound is obtained from primes in disjoint logarithmic intervals using a fourth-moment argument based on the multiplicative large sieve and a subset-product second moment. 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