Which of the following will leave the largest reminder upon division by 7?

So $2019 \pmod {1010} \equiv 1009$ and .... oops your answer requires we divide by less than or equal to $1000$.

Okay if we divide $N$ by $\lfloor \frac Nk \rfloor + 1$ we get....

.... well if $\lfloor \frac Nk \rfloor = q$ if and only if $qk \le N < (q+1)k$.

So if we divide by $q+1$ we get $(k-1)(q+1) =kq -q+k-1$ and if $q > k$ (in other words of $k< \sqrt N$) then $(k-1)(q+1)=kq-q+k -1 < qk \le N < (q+1)k$ so

the remainder will be $N-(k-1)(q+1) > qk -(kq-q+k-1)=q-k+1$.

We can maximize that by make $k$ really small and $q$ really big.

If we make $k =2$, and $q = 1010$ as I did at the top of the post that is the biggest remainder. But $q=1010$ is to big to divide by. So let $k =2$. $3\times 673 = 2019$ so let $q = 674$.

THen $2\times 674 < 2019 < 3\times 674$ and the remainder is $2019 - 2\times 674 =671$.

For any $n=673$ or $672$ you get remainders $0$ and $3$ when dividing by $n$. $2019 = 3\times 673$ and $2019 = 3\times 672 + 3$. For $n \le 671$ you get a remainder less than $n$ and so less than $671$.

And for any $n = 674 + m$ where $1\le m \le 1000-673= 327$ we have $2(673 + m) < 2000 < 2019 < 3(673+m) = 2019 +m$ so the remainder is $2019 - 2(674+m) = 671 - 2m < 671$.

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The number of American households that were unbanked last year dropped to its lowest level since 2009, a dip due in part to people opening accounts to receive financial assistance during the pandemic, a new report says.  

Roughly 4.5% of U.S. households – or 5.9 million – didn't have a checking or savings account with a bank or credit union in 2021, a record low, according to the Federal Deposit Insurance Corporation's most recent survey of unbanked and underbanked households. 

Roughly 45% of households that received a stimulus payment, jobless benefits or other government assistance after the start of the pandemic in March, 2020 said those funds helped compel them to open an account, according to the biennial report which has been conducted since 2009.

"Safe and affordable bank accounts provide a way to bring more Americans into the banking system and will continue to play an important role in advancing economic inclusion for all Americans,'' FDIC acting chairman Martin J. Gruenberg said in a statement.  

A lack of banking options delayed some households from getting federal payments aimed at helping the country weather the economic fallout from the COVID-19 health crisis.

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The FDIC initiated an educational campaign to get more Americans to open an account to enable the direct deposit of those funds. And banks such as Capital One and Ally Financial ended  overdraft and other fees that have been a key barrier to some Americans accessing the banking system. 

What does it mean to be unbanked?

A household is deemed unbanked when no one in the home has an account with a bank or credit union. That share of households has dropped by nearly half since 2009. And since 2011, when 8% of U.S. households were unbanked, the highest since the start of the survey, and the record low reached in 2021, roughly half of the drop was due to a shift in the financial circumstances of American households the FDIC says.

Who are the underbanked?

Those who have a checking or savings account, but also use financial alternatives like check cashing services are considered underbanked. The underbanked represented 14% of U.S. households, or 18.7 million, last year.   

Why are people unbanked or underbanked?

Many of those who are unbanked say they can't afford to have an account because of the fees for insufficient funds and overdrafts that are tacked on when account balances fall short. Roughly 29% said fees or not having the required minimum balance were the primary reasons they didn't have a checking or savings account, as compared to 38% who cited those obstacles in 2019.

Are some groups more likely to be unbanked? 

The numbers of the unbanked were greater among households that included those who were working age and disabled, lower income, included a single mother, or were Black or Hispanic. Among white households for instance, 2% didn't have a bank account last year as compared to 11% and 9% of their Black and Hispanic counterparts.

Meanwhile, nearly 15% of households with a working age member who had a disability were unbanked compared to almost 4% of other households. And  nearly 16% of households with a single mother were unbanked as compared to about 2% of married couples who lacked an account. 

 "These gaps attest there's still a lot of opportunity to expand participation across the population in the banking system,'' Keith Ernst, Associate Director of Consumer Research and Examination Analytics at the FDIC, said during a media call about the report.            

Will the number of unbanked rise if the U.S. has a recession? 

Perhaps.

"During the last recession unbanked rates did indeed go up,'' Karyen Chu, chief of the Banking Research Section at the Center for Financial Research, said during the call. 

Additionally, last year, homes where the head of household was out of work were nearly five times more likely to not have a bank account as compared to those where the household head was employed.

Recession predictions 2023:Is a recession inevitable in 2023? Here's what experts are saying.

"To the extent that income goes down ... that has generally been associated with increases in unbanked rates,’’ Chu said. 

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