Practice (111)

The positive integers

$N$ and

$N^2$ both end in the same sequence of four digits

$abcd$ when written in base 10, where digit

$a$ is not zero. Find the three-digit number

$abc$ .

272

What is the last digit of

$9^{2019}$ ?

273

What are the last two digits of

$8^{88}$ ?

274

Find the remainder when

$3^{2019} + 4^{2019}$ is divided by 5?

280

Find the remainder when

$9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999}}$ is divided by

$1000$ .

411

What is the units digit of the sum of the squares of the integers from

$1$ to

$2015$ , inclusive?

1117

What are the last two digits in the sum of the factorials of the first

$100$ positive integers?

1171

What is the tens digit of

$7^{2019}$ ?

1186

What is the units digit of

$13^{2019}$ ?

1259

What are the sign and units digit of the product of all the odd negative integers strictly greater than

$-2015$ ?

1482

What is the hundreds digit of

$2011^{2011}?$

1508

The number obtained from the last two non-zero digits of

$90!$ is equal to

$n$ . What is

$n$ ?

1608

Let

$k={2008}^{2}+{2}^{2008}$ . What is the units digit of

$k^2+2^k$ ?

1721

What is the tens digit in the sum

$7!+8!+9!+...+2018!$

1819

What is the units digit of the product

$7^{23} \times 8^{105} \times 3^{18}$ ?

1861

When

$(37 \times 45) - 15$ is simplified, what is the units digit?

2114

Find the smallest positive integer

$n$ such that the last

$3$ digits of

$n^3$ is

$888$ .

2362

Find all positive integer

$n$ such that

$n$ is a square and its last four digits are the same.

2540

What is the last digit of

$17^{17^{17^{17}}}$ ?

2621

If for any integer

$k\ne 27$ and

$\big(a-k^{2015}\big)$ is divisible by

$(27-k)$ , what is the last two digits of

$a$ ?

2654

Determine the units digit of the sum

$0!+1!+2!+\cdots+n!+\cdots+20!$ ?

2659

What is the units digit of

$-1\times 2008 + 2 \times 2007 - 3\times 2006 + 4\times 2005 +\cdots-1003\times 1006 + 1004 \times 1005$ ?

2741

What is the last digit of

$7^{222}$ ?

2913

What is the tens digit of $2015^{2016}-2017?$

4157

What is the tens digit of

$321^{123}$ ?

Root SquareNumber EndingDigits

AMC10/12 AIME PUMaC 华杯

US China

With Solutions

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