Solving Divisibility Problems: Number 94x29y6

The problem asks for the value of $(2x + 3y)$ for a seven-digit number 94x29y6 that is divisible by 72, with the condition that $x \ne y$.

A number is divisible by 72 if and only if it is divisible by both 8 and 9, because 8 and 9 are coprime factors of 72.

Divisibility Rule for 8

A number is divisible by 8 if the number formed by its last three digits is divisible by 8.

In the number 94x29y6, the last three digits are 9y6. So, 9y6 must be divisible by 8.

Let's check possible values for the digit y (from 0 to 9):

• If y=0, 906 ÷ 8 = 113.25 (Not divisible)
• If y=1, 916 ÷ 8 = 114.5 (Not divisible)
• If y=2, 926 ÷ 8 = 115.75 (Not divisible)
• If y=3, 936 ÷ 8 = 117 (Divisible)
• If y=4, 946 ÷ 8 = 118.25 (Not divisible)
• If y=5, 956 ÷ 8 = 119.5 (Not divisible)
• If y=6, 966 ÷ 8 = 120.75 (Not divisible)
• If y=7, 976 ÷ 8 = 122 (Divisible)
• If y=8, 986 ÷ 8 = 123.25 (Not divisible)
• If y=9, 996 ÷ 8 = 124.5 (Not divisible)

From this analysis, the possible values for y are 3 or 7.

Divisibility Rule for 9

A number is divisible by 9 if the sum of its digits is divisible by 9.

For the number 94x29y6, the sum of the digits is:

Sum = $9 + 4 + x + 2 + 9 + y + 6 = 30 + x + y$

For the number to be divisible by 9, the sum $(30 + x + y)$ must be a multiple of 9.

Since x and y are digits (0-9), the maximum value of $x+y$ is $9+9=18$. The minimum value is $0+0=0$.

So, the sum $30+x+y$ must be between $30+0=30$ and $30+18=48$.

The multiples of 9 between 30 and 48 are 36 and 45.

So, $30 + x + y$ must be either 36 or 45.

Combining Divisibility Conditions

Now we combine the possible values for y (3 or 7) with the possible sums of digits (36 or 45).

Case 1: y = 3

The sum of digits is $30 + x + 3 = 33 + x$.

This sum must be a multiple of 9. The next multiple of 9 after 33 is 36.

So, $33 + x = 36$. This gives $x = 36 - 33 = 3$.

In this case, $x=3$ and $y=3$. However, the problem states that $x \ne y$. So, this case is not valid.

The next multiple of 9 is 45. $33+x=45 \implies x=12$, which is not a single digit.

Case 2: y = 7

The sum of digits is $30 + x + 7 = 37 + x$.

This sum must be a multiple of 9. The next multiple of 9 after 37 is 45.

So, $37 + x = 45$. This gives $x = 45 - 37 = 8$.

In this case, $x=8$ and $y=7$. This satisfies the condition $x \ne y$ (since $8 \ne 7$).

The next multiple of 9 is 54. $37+x=54 \implies x=17$, which is not a single digit.

Therefore, the only valid values for x and y are $x=8$ and $y=7$. The number is 9482976. Let's quickly check: 9482976 / 8 = 1185372, 9482976 / 9 = 1053664. It is divisible by both 8 and 9, thus divisible by 72.

Calculating the Value of (2x + 3y)

We need to find the value of $(2x + 3y)$ using $x=8$ and $y=7$.

$2x + 3y = 2(8) + 3(7)$

$2(8) = 16$

$3(7) = 21$

$2x + 3y = 16 + 21 = 37$

The value of $(2x + 3y)$ is 37.

Revision Table: Divisibility Rules

Additional Information on Number Divisibility

Divisibility rules are useful shortcuts to determine if a number is exactly divisible by another number without performing the actual division. Understanding these rules is crucial for solving problems involving factors and multiples.

• When a number is divisible by a composite number (like 72), it must be divisible by all the prime factors of that number raised to their highest power in its prime factorization. For 72, $72 = 8 \times 9 = 2^3 \times 3^2$. So, the number must be divisible by $2^3=8$ and $3^2=9$. Since 8 and 9 are coprime, we can use the individual rules for 8 and 9.
• If the composite number is a product of two coprime numbers, n and m, the number is divisible by their product (n × m) if and only if it is divisible by both n and m individually. For example, divisibility by 30 means divisibility by both 3 and 10 (since gcd(3,10)=1). Divisibility by 12 means divisibility by both 3 and 4 (since gcd(3,4)=1), but not necessarily by 2 and 6 (since gcd(2,6)≠1).
• Always consider the constraints given in the problem, such as $x \ne y$ in this question, as they help eliminate possibilities.