UKMT Special (Problem $15$ )

The five-digit number $a679b$ is a multiple of $72$

What are the values of $a$ and $b$ ?

[UKMT Hamilton Olympiad $2015$ , H $1$ ]

#Algebra

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
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Comments

You have until next Monday, $3:00$ pm!

Yajat Shamji - 6 months, 4 weeks ago

Step 1) A number is divisible by 72, if and only if it is divisible by 8 and 9.

Step 2) According to divisibility rules, a679b is divisible by 8 if 79b is divisible by 8.

   Only b = 2 that satisfy.

Step 3) According to divisibility rules, a6792 is divisible by 8 if a + 6 + 7 + 9 + 2 is divisible by 9.

   a + 6 + 7 + 9 + 2 = a + 24 is divisible by 9.
   Only a = 3 that satisfy

The five-digit number is 36792.

Quest Keeper - 6 months, 3 weeks ago

Correct solution and method!

Yajat Shamji - 6 months, 3 weeks ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

UKMT Special (Problem 151515)

Comments

UKMT Special (Problem $15$ )